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B   M   aSD   Mflb 


OL  ALGEBRA 

FIRST  COURSE 


H1ETZ,CRATH0RNE,AND  TAYLOR 


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HENRY         HOLT         AND         COMPANY 

NEW   YORK  CHICAGO 


SCHOOL    ALGEBRA 

FIRST  COURSE 


BY 

H.  L.  RIETZ,  Ph.D. 

UNIVERSITY   OF   ILLINOIS 

A.  R.  CRATHORNE,  Ph.D. 

UNIVERSITY  OF   ILLINOIS 


E.  H.  TAYLOR,  Ph.D. 

EASTERN   ILLINOIS   STATE    NORMAL   SCHOOL 


NEW  YORK 

HENRY  HOLT  AND  COMPANY 

1915 


Copyright,    1915 

BY 
HENRY    HOLT    AND    COMPANY 


?N/\i54- 


PREFACE 

This  book  is  the  first  volume  of  a  two-book  series.  It  eon- 
tains  ample  material  for  a  full  year's  work  in  the  first  year  of  the 
high  school,  and  covers  the  parts  of  algebra  most  likely  to  be 
of  use  to  the  student  who  goes  no  further  in  the  subject. 
It  will  prepare  for  Plane  Geometry  and  Physics,  which  come 
in  the  later  years  of  the  high  school.  The  second  volume, 
the  Advanced  Course,  supplies  the  additional  material  de- 
manded for  entrance  into  the  scientific  and  technical  courses 
in  our  colleges  and  universities. 

The  text  represents  a  special  effort  at  presentation  of  prin- 
ciples and  definitions  in  clear,  simple  style,  with  wordy  mies- 
sentials  eliminated.  The  study  of  algebra  is  taken  up  as  an  ex- 
tension of  arithmetic.  The  laws  of  algebra  are  first  suggested 
by  induction  from  familiar  rules  of  arithmetic,  and  throughout 
the  book  the  close  connection  with,  arithmetic  is  kept  in  view. 
The  pupil  is  led  to  see  that  new  symbols  are  introduced  into 
algebra,  not  arbitrarily,  but  because  of  their  real  advantages 
in  representing  numbers. 

Difficulties  are  taken  one  at  a  time  at  sufficient  intervals  to 
allow  the  mastery  of  each  one  before  proceeding  to  the  next. 
Thus  the  fundamental  idea  of  representmg  numbers  by  letters 
is  developed  in  the  first  two  chapters;  that  of  signed  numbers  in 
Chapter  III;  while  the  equation  is  not  formally  introduced 
until  Chapter  IV,  though  it  is  used  informally  without  special 
definition  from  the  beginning.  In  accordance  with  the  same 
general  scheme,  each  of  the  several  chapters  on  the  equation 
takes  up  a  single  now  difficulty  at  a  time  when  the  pupil  is 
ready  for  it. 


iv  PREFACE 

A  fourth  fundamental  topic,  the  notion  of  a  function,  is  gradu- 
ally approached  through  evaluation  of  expressions  and  through 
the  graph,  so  that  the  pupil  is  prepared  for  the  more  formal  treat- 
ment, with  the  use  of  the  functional  notation,  in  the  advanced 
course.  Graphical  work  is  treated  as  simply  as  possible,  not 
as  an  added  difficulty,  but  when  occasion  for  it  arises  in  the 
algebraic  work.  While  graphical  solution  is  important,  it  is 
not  the  fundamental  notion  imderlying  graphical  representa- 
tion. The  idea  of  functionality,  the  change  in  a  function  as  the 
independent  variable  changes,  is  the  idea  about  which  the 
work  in  graphs  should  center. 

The  book  is  well  supplied  with  carefully  graded  exercises 
and  problems.  In  general,  preference  has  been  given  to  mis- 
cellaneous groups  rather  than  to  groups  illustrating  but  one 
process.  With  regard  to  the  question  of  correlation  with  other 
high  school  subjects  the  book  follows  a  middle  course.  Many 
problems  involving  applications  to  other  subjects  have  been 
introduced,  but  care  has  been  taken  to  keep  within  the  limits 
of  the  pupil's  experience,  as  well  as  to  give  data  that  may  be 
depended  upon  to  be  correct. 

During  the  last  decade  many  teachers'  associations  have  dis- 
cussed the  arrangement  and  content  of  the  high  school  course 
in  algebra.  Many  outlines  of  courses  have  been  proposed  and 
some  detailed  syllabi  have  been  published.  In  the  preparation 
of  this  book  we  have  availed  ourselves  of  these  discussions  and 
printed  syllabi,  and  have  endeavored  to  incorporate  into  the 
work  the  views  which  prevail  among  progressive  teachers.  We 
take  pleasure  in  expressing  to  many  teachers  our  appreciation 
for  helpful  suggestions  and  criticisms.  We  are  especially  in- 
debted to  Prof.  E.  J.  Townsend  and  Dr.  E.  B.  Lytle  of  the  Uni- 
versity of  Illinois  and  to  Miss  Jessie  D.  Brakensiek  of  Quincj'' 
High  School,  Quincy,  111.,  for  careful  and  critical  reading 
of  the  manuscript  and  for  suggestions  as  to  exercises  and 
problems;  to  Mr.  J.  L.  Dunn  of  the  Lewis  and  Clark  High 
School,  Spokane,  Wash.,  Mr.  C.  H.  Fullerton  and  Mr.  W.  B. 


PREFACE  V 

Skimming  of  the  East  High  Scliool,  Columbus,  O.,  Mr.  E.  A. 
Hook  of  the  Commercial  High  School,  Brooklyn,  N.Y.,  Mr. 
L.  C.  Irwin  of  the  Jolict  Township  High  School,  Jolict,  111., 
and  Mr.  R.  L.  Modesitt  of  the  Eastern  Illinois  State  Normal 
School,  Charleston,  111.,  for  reading  the  proof  and  seeing  the 
book  through  the  press. 

H.    L.    RIETZ 

A.    R.    CRATHORNE 

E.    H.    TAYLOR 


CONTENTS 

CHAPTER   I 
INTRODUCTION 

PAGE 

1.  Numbers  and  Language  of  Arithmetic 1 

2.  Language  of  Algebra 1 

3.  Symbols  of  Operation 1 

4.  Use  of  Letters  in  Solving  Problems 4 

5.  Use  of  Letters  to  Abbreviate  Statements 6 

6.  Area  of  a  Triangle 7 

7.  Circumference  of  a  Circle 7 

8.  Area  of  a  Circle 7 

9.  Factors 9 

10.  Coefficient 9 

11.  Exponents  and  Powers 9 

12.  Historical  Note  on  Symbols 12 

CHAPTER   n 
ALGEBRAIC    EXPRESSIONS 

13.  Algebraic  Expressions 14 

14.  Order  of  Operations 14 

15.  Use  of  Parentheses lo 

16.  Evaluation  of  Expressions 16 

17.  Expressions  Containing  one  Letter 17 

18.  Graphical  Representation 19 

CHAPTER   III 
POSITIVE  AND   NEGATIVE   NUMBERS 

19.  The  LTse  of  a  Scale  to  Represent  the  Numbers  of  Arithmetic  .    .  2.3 

20.  Addition  and  Subtraction  on  the  Scale 2.3 

21.  Positive  and  Negative  Numbers 24 


viii  CONTENTS 

PAGE 

22.  Illustrations 25 

23.  Numerical  or  Absolute  Value 25 

24.  Greater  and  Less 25 

25.  Addition  of  Signed  Numbers 28 

26.  Subtraction  of  Signed  Numbers 29 

27.  Subtraction  on  a  Scale 30 

28.  Rule  for  Subtraction 31 

29.  Addition  and  Subtraction  of  Several  Numbers 34 

30.  Multiplication  in  Arithmetic 35 

31.  Multiplication  of  Signed  Numbers 35 

32.  Division  of  Signed  Numbers 38 

33.  Fractions 39 

CHAPTER   IV 
EQUALITIES 

34.  Members  of  an  Equality 42 

35.  Identities 42 

36.  Equations 43 

37.  Solution  of  Equations 44 

38.  Principles  Used  in  Solving  Equations 44 

39.  Verification  of  Solutions  by  Substitution 45 

40.  Transposition 46 

41.  Translation  of  English  Expressions  into  Algebraic  Expressions  .  47 

CHAPTER  V 
ADDITION 

42.  Terms  of  an  Expression 53 

43.  Monomials  and  Polynomials 53 

44.  Similar  Terms 53 

45.  Addition  of  Monomials 54 

46.  Simplifying  Polynomials 56 

47.  Arrangement  of  Terms  in  a  Polynomial 57 

48.  Addition  of  Polynomials 57 

CHAPTER  VI 
SUBTRACTION 

49.  Subtraction  of  Monomials 60 

50.  Subtraction  of  Polynomials 61 


CONTENTS  ix 

CHAPTER  VII 
PARENTHESES 

PAGE 

51.  Removal  of  Parentheses 64 

52.  Insertion  of  Parentheses 66 

53.  Collecting  Literal  Coefficients 68 

CHAPTER  Vni 
MULTIPLICATION 

54.  Products  of  Powers 69 

55.  Products  of  Monomials 70 

56.  Multiplication  of  a  Product  by  any  Number 72 

57.  Product  of  a  Polynomial  by  a  Monomial 72 

58.  Product  of  Two  Polynomials 75 

CHAPTER   IX 
EQUATIONS    AND   PROBLEMS 

59.  Equations  Involving  Parentheses 78 

60.  Equations  Involving  Fractions 81 

CHAPTER  X 
DIVISION 

61.  Division  of  Monomials 86 

62.  Division  of  a  Polynomial  by  a  Monomial 87 

63.  Division  by  a  Polynomial 89 

64.  Literal  Coefficients       92 

CHAPTER   XI 
LINEAR  EQUATIONS 

65.  Linear  Equations 96 

CHAPTER   XII 
IMPORTANT  TYPE   FORMS 

66.  Introductory 100 

67.  Square  of  a  Binomial 100 


X  CONTENTS 

PAGE 

68.  Product  of  the  Sum  and  Difference  of  Two  Numbers 102 

69.  Product  of  Two  Binomials  Having  a  Common  Term 104 

70.  Cube  of  a  Binomial 105 

71.  Square  of  a  Trinomial 106 

CHAPTER   Xni 
FACTORING 

72.  Prime  Factors  in  Arithmetic 109 

73.  Prime  Factors  in  Algebra 109 

74.  Factors  of  Monomials 110 

75.  Monomial  Factors  in  Polynomials Ill 

76.  Factors  Found  by  Grouping  Terms 112 

77.  Difference  of  Two  Squares 113 

78.  Trinomial  Squares 114 

79.  Trinomials  of  the  Form  ^2  +  (a  +  6)  X  +  a6 116 

80.  General  Quadratic  Trinomial 118 

81.  Sum  and  Difference  of  Two  Cubes 119 

82.  Summary  of  Factoring 121 

CHAPTER   XIV 
EQUATIONS   SOLVED   BY  FACTORING 

83.  Quadratic  Equations 126 

84.  Factoring  AppUed  to  the  Solution  of  Quadratic  Equations  .    .    .  127 

CHAPTER  XV 

HIGHEST    COMMON   FACTOR  AND   LOWEST   COMMON 
MULTIPLE 

85.  Greatest  Common  Divisor  in  Arithmetic 1.32 

86.  Highest  Common  Factor 132 

87.  Lowest  Common  Multiple 134 

CHAPTER   XVI 
FRACTIONS 

88.  Fractions  in  Arithmetic 136 

89.  Fractions  in  Algebra 136 

90.  Division  ])y  Zero 138 


CONTENTS  xi 

PAGK 

91.  Signs  in  Fractions 138 

92.  Reduction  to  Lowest  Terms 140 

93.  Cancellation 141 

94.  Reduction  to  Common  Denominator 142 

95.  Addition  and  Subtraction  of  Fractions 143 

96.  Multiplication  of  Fractions 145 

97.  Division  of  Fractions 149 

98.  Complex  Fractioas 151 

CHAPTER   XVII 
FRACTIONAL  AND   LITERAL  EQUATIONS 

99.    Clearing  Equations  of  Fractions 159 

100.  Unknowns  in  the  Denominator 160 

101.  Literal  Equations 164 

102.  Subscript  Notation 165 

CHAPTER  XVIII 
RATIO,  PROPORTION,   AND   VARIATION 

103.  Ratio 169 

104.  Proportion 170 

105.  Mean  Proportional 171 

106.  Third  and  Fourth  Proportional 171 

107.  Proportion  by  Alternation 173 

108.  Proportion  by  Inversion 173 

109.  Proportion  by  Composition 173 

110.  Proportion  by  Division 174 

111.  Proportion  by  Composition  and  Division 174 

112.  Variables  and  Constants 175 

^113.   Direct  Variation 175 

CHAPTER  XIX 

GRAPHICAL   REPRESENTATION    OF   THE   RELATION 
BETWEEN   TWO    VARIABLES 

114.  Introduction 180 

115.  Axes,  Coordinates ISO 

116.  Plotting  of  Points 182 

117.  Use  of  Coordinate  Paper 182 

118.  Variables 183 

119.  Definition  of  a  Function 183 


xii  CONTENTS 

PAGE 

120.  Graph  of  a  Function 184 

121.  Graph  of  an  Equation 186 

122.  Locus  of  a  Linear  Equation 187 

123.  Graphic  Solution  of  Equations 187 

124.  Grapliical  Representation  of  Scientific  Data 189 

CHAPTER  XX 
SYSTEMS   OF   LINEAR  EQUATIONS 

125.  Solution  of  Equations  in  Two  or  More  Unknowns 192 

126.  Simultaneous  Equations 193 

127.  Independent  Equations 193 

128.  Dependent  or  Equivalent  Equations 193 

129.  Inconsistent  Equations 193 

130.  Elimination 194 

131.  Elimination  by  Addition  and  Subtraction 195 

132.  Elimination  by  Substitution 197 

133.  Standard  Form  ax  +  by  +  c  =  o 199 

134.  Literal  Equations  Containing  Two  Unknowns 205 

135.  Linear  Systems  in  Three  or  More  Unknowns 206 

CHAPTER  XXI 
SQUARE   ROOT   AND   APPLICATIONS 

136.  Definition  of  a  Square  Root 214 

137.  Radical  Sign 214 

138.  Square  Root  of  Monomials 214 

139.  Equations  Solved  by  Finding  Square  Roots 215 

140.  Square  Roots  of  Trinomials 216 

141.  Process  of  Finding  the  Square  Root 216 

142.  Square  Roots  of  Numbers  Expressed  in  Arabic  Figures  ....  219 

143.  Explanation  of  Process  of  Finding  Square  Root  in  Arithmetic  .  220 

144.  Number  with  More  than  Two  Periods 220 

145.  Square  Roots  of  Decimals 221 

146.  Approximate  Square  Root 222 

CHAPTER  XXII 
RADICALS 

147.  Radicals 226 

148.  Rational  and  Irrational  Numbers 226 


CONTENTS  xiii 

PAGK 

149.  Surds 22G 

150.  Square  Root  of  a  Fraction  — 227 

b 

Simplification  of  Radicals 227 

Meaning  of  Simplification  of  a  Radical 228 

Addition  and  Subtraction  of  Radicals 229 

Multiplication  of  Quadratic  Surds 231 

Division  of  Quadratic  Surds 232 

Rationalization  of  Denominators 233 

Rationalizing  Factor 234 

Solution  of  Equations  Involving  Radicals 235 


CHAPTER  XXIII 
QUADRATIC   EQUATIONS 

Quadratic  Equations  Solved  by  Factoring 238 

Completing  the  Square 239 

Equations  Solved  by  Completing  Square 239 

Solution  by  Hindu  Method  of  Completing  Square 240 

Type  Form  of  a  Quadratic  Equation 242 

Quadratic  Solved  by  Formula 242 

The  Special  Quadratic  ax-  +  c  =  o 245 

Graphs  of  Quadratic  Functions 247 

Imaginary  Numbers 249 

Graphical  Meaning  of  Imaginary  Roots 250 

Historical  Note  on  Quadratics 255 

CHAPTER  XXIV 
SYSTEMS   OF  EQUATIONS  INVOLVING    QUADRATICS 

Introduction 25G 

Solution  of  Simultaneous  Quadratics 257 

One  Equation  Linear  and  One  Quadratic 257 

Equations  Containing  x^  and  y"^  only 259 

Special  Methods 2G0 

Index 267 


SCHOOL   ALGEBEA 

FIRST  COURSE 
CHAPTER  I 

INTRODUCTION 

1.  Numbers  and  language  of  arithmetic.  In  counting,  the 
child  learns  numbers  which  are  called  integers. 

The  written  language  of  arithmetic  uses  the  numerals  0,  1, 
2,  3,  4,  5,  6,  7,  8,  9,  to  represent  numbers  and  the  signs  +,  -, 
X,  -T-  to  denote  operations. 

In  problems  where  two  or  more  integers  are  added  or  are 
multiplied  together,  or  when  the  smaller  of  two  integers  is  sub- 
tracted from  the  greater,  the  answer  is  always  an  integer.  In 
the  use  of  the  sign  ^  however,  another  kind  of  numbers,  called 
fractions,  is  obtained.  These  two  kinds  of  numbers,  integers 
and  fractions,  have  been  studied  in  arithmetic. 

2.  Language  of  algebra.  Algebra  is  a  continuation  of  arith- 
metic. In  it  we  use  not  only  the  numbers  and  symbols  of 
arithmetic,  but  we  also  introduce  new  kinds  of  numbers  and 
new  symbols.  The  written  language  of  algebra  makes  much 
use  of  letters  to  represent  numbers.  This  use  of  letters  is  not 
entirely  new  to  the  student,  for  it  is  customary  in  arithmetic  to 
represent  certain  numbers  by  letters. 

Thus,  the  radius  of  a  circle  is  often  represented  by  r,  the  diameter  by 
d,  or  by  2  X  r.  The  altitude  and  the  base  of  a  rectangle  may  be  repre- 
sented by  a  and  b,  the  area  by  a  Xb,  and  the  distance  around  by 
a  +  a  +  b  +  b,  or  2xa  +  2xb. 

3.  Symbols  of  operation.  The  signs  +,  -,  x,  -^  are  used 
in  algebra  as  in  arithmetic,  Ijut  the  sign  of  multiplication  is 


2  INTRODUCTION  [Chap.  I. 

usually  omitted.  For  example,  a  x  6  is  usually  written  ab, 
and  2  X  r  is  written  2r.  If  a  sign  of  multiplication  is  used,  it  is 
customary  to  use  a  dot,  written  a  little  above  the  position  for 
a  period  to  distinguish  it  from  the  decimal  point,  instead  of 
the  sign  X.  For  example,  2  x  3  is  written  2  •  3.  The  sign  of 
division  is  not  much  used  in  algebra.     Thus,  a  -=-  6  is  usually 

written  -r- 

0 

The  use  of  letters  to  represent  numbers  enables  us  to  write 
many  statements  in  very  brief  form.  Thus,  if  A  is  the  area,  6 
the  base,  and  a  the  altitude  of  a  rectangle,  the  brief  statement 

A  =  ab 

gives  the  rule  that  the  area  of  a  rectangle  is  equal  to  the  prod- 
uct of  the  base  and  altitude. 

It  is  important  to  be  able  to  translate  English  sentences  into 
such  algebraic  statements. 

EXERCISES 

1.  If  I  stands  for  the  length  of  a  running  track  in  yards, 
what  stands  for  the  length  of  a  track  50  yards  longer?  Ans. 
I  +  50  yards. 

2.  If  I  stands  for  the  length  of  a  track,  what  is  the  length 
of  a  track  100  yards  longer?  What  is  the  length  of  a  track 
twice  as  long? 

3.  What  is  the  cost  of  5  railway  tickets  at  x  dollars  a 
ticket? 

4.  What  is  the  value  of  3a  when  a  is  1?  When  a  is  2? 
When  a  is  13? 

5.  How  many  pecks  are  there  in  6  bushels?     In  x  bushels? 

6.  How  many  inches  are  there  in  y  feet? 

7.  If  n  represents  a  number,  what  represents  a  number 
three  times  as  large? 

8.  Write  the  sum  of  a,  b,  c,  and  d,  using  the  sign  of  addition. 

9.  Write  the  product  of  a  and  x  as  it  is  expressed  in  algebra. 


Art.  3]  LANGUAGE   OF  ALGEBRA  3 

10.  Why  is  it  not  good  form  to  write  25  for  2x5? 

11.  Write  the  following  without  using  a  sign  of  multiplica- 
tion :     8  X  a:,  6  X  6,  7  X  a  X  6,  a:  X  y  X  z,  5  X  r  X  ^  9  X  m  X  n  X  y. 

12.  Indicate  the  subtraction  of  x  from  a  by  using  the  sign 
of  subtraction. 

13.  Write  the  product  of  a,  b,  c,  and  d  in  three  different  ways. 

14.  If  a  is  an  integer,  what  is  the  next  integer?  What  is 
the  preceding  integer? 

15.  A  barrel  contains  31|  gallons.  If  c  is  the  number  of 
cubic  inches  in  a  gallon,  what  is  the  number  of  cubic  inches 
in  a  barrel? 

16.  If  X  represents  a  certain  number,  what  represents  a 
number  10  greater  than  twice  x? 

17.  If  p  is  the  cost  of  one  article,  what  is  the  cost  per 
dozen? 

18.  If  d  is  the  cost  per  dozen,  what  is  the  cost  of  one? 

19.  What  is  the  simple  interest  on  $100  for  three  years  at 
6  %  per  annum?     On  x  dollars  for  y  years  at  m  per  cent? 

20.  If  x  and  y  are  the  lengths  of  two  lines,  what  is  their 
combined  length? 

21.  The  age  of  a  father  is  30  years  more  than  twice  his 
son's  age.     If  x  is  the  son's  age,  what  is  the  father's  age? 

22.  A  rectangle  is  twice  as  long  as  it  is  wide.  Let  x  be  its 
width.     What  is  its  length?    Its  perimeter?    Its  area? 

23.  A  rectangle  is  8  feet  wide.  If  x  is  the  length,  what  is 
the  area;   the  perimeter? 

24.  Write  in  algebraic  language  the  statement  that  the 
sum  of  two  numbers,  a  and  b,  is  divided  by  the  number  c. 

25.  Five  times  6  plus  three  times  6  plus  two  times  6  equals 
how  many  6's? 

26.  5x  +  3x  +  2x  =  how  many  x's? 

27.  6a  +  2a  +  a  =  ? 

28.  8x-^x  +  2x  +  5x=  ? 

29.  y  +  5y-  Sy  +  ^^  =  ? 

30.  3z+lz-  z+iz+Qz-lz=  ? 


4  INTRODUCTION  [Chap.  L 

Fill  out  the  blanks  in  the  following  : 

31.  16  +  56  -  46  -  6  =  (  )  6. 

32.  dx+  (  )  X-  5x=  Qx. 

33.  (  )  a  +  3a  -  2rt  -  |a  =  4a. 

4.  Use  of  letters  in  solving  problems.  The  following 
examples  show  the  further  use  of  letters  to  represent  numbers 
and  to  simplify  the  solution  of  certain  problems. 

Example  1.  The  sum  of  two  numbers  is  128.  The  larger  is  3  times 
the  smaller.     Find  the  numbers. 

Let  n  =  the  smaller  number. 
Then  3n  =  the  larger  number, 

and  71  +  3n  =  128,  the  sum. 

Adding,  in  =  128. 

Dividing  by  4,       n  =  32,  the  smaller  number. 
Then  3n  =  96,  the  larger  number. 

Example  2.  A  lot  is  sold  for  $720,  which  is  20%  more  than  it  cost. 
What  was  the  cost? 

The  following  is  a  common  form  of  the  arithmetical  solution. 

Let  100%  of  the  cost  =  the  cost. 
Then  120%  of  the  cost  =  $720, 

1%  of  the  cost  =  xis  of  $720  =  $6, 
and  100%  of  the  cost  =  100  x  $6  =  $600. 

Hence,  the  lot  cost  $600. 

The  solution  may  be  shortened  by  the  use  of  letters. 
Let  c  =  the  number  of  dollars  the  lot  cost. 
Then  1.20c  =  720, 

and  c  =  720  ^  1.20  =  600. 

Hence,  the  lot  cost  $600. 

Note  that  c  is  a  number  and  is  not  the  cost. 


EXERCISES   AND   PROBLEMS 

1.  A  house  and  lot  are  worth  $6000.  The  house  is  worth 
four  times  as  much  as  the  lot.     What  is  the  value  of  the  lot? 

2.  During  a  January  cold  wave  the  price  of  soft  coal  in- 
creased 10%.  At  the  end  of  the  cold  period  the  price  was 
$3.85  per  ton.     What  was  the  price  at  the  beginning? 


Akt.  4]    USE  OF  LETTERS  IN  PROBLEMS         5 

3.  A  book  sells  for  S2.40.  The  dealer  makes  a  profit  of 
25'^,  of  the  cost  price.  What  was  the  cost  of  the  book  to 
the  dealer? 

4.  The  sum  of  the  edges  of  a  cube  is  48  inches.  What  is 
the  length  of  one  edge? 

5.  A  merchant  sells  a  hat  for  $3.50,  making  25  %  profit. 
What  did  the  hat  cost  the  merchant? 

6.  For  what  number  does  a;  stand  in  the  equality  3  +  a:  =  13? 
Hint:  Here  x  is  the  number  which  added  to  3  gives  13. 

In  the  following  exercises,  the  letters  stand  for  unknown 
numbers.     Find  the  numbers. 


7. 

a:  -  2  =  5. 

8. 

x+  10=  12. 

9. 

2z  =  24. 

10. 

4.r  =  28. 

11. 

2-5=0. 

12. 

52  =  10. 

13. 

7y  +  5y=  120 

14. 

Sw  ^  21. 

15. 

.r  -  20  =  90. 

16. 

12n  -  5n+n=  64. 

17. 

x  +  30=  75. 

18. 

Sy  =  63. 

19. 

w-S^  11. 

20. 

.r  +  5  =  8  +  2. 

21. 

z+2z+3z=  12. 

)le 

t;   add  15.     If  the 

result 

22.  Thmk  of  a  number; 
is  52,  what  was  the  first  number? 

23.  If  a  certain  num})er  is  multiplied  by  4,  the  result  is  5 
more  than  33.     What  is  the  number? 

24.  A  boy  bought  3  books.  One  book  cost  twice  as  much 
as  the  other  two  put  together.  If  these  two  books  were  the 
same  price,  and  if  he  received  4  cents  change  out  of  a  dollar, 
what  was  the  price  of  each  book? 

25.  Two  men,  Smith  and  Jones,  do  a  certain  piece 
of  work  for  $300  at  $2.00  per  day  each.  Smith  works  5 
times  as  many  days  as  Jones.  How  many  days  did  each 
work? 

26.  In  a  fire,  Smith  lost  3  times  as  much  as  Jones,  and 
Brown  lost  4  times  as  much  as  Jones.  If  the  combined  loss 
was  $5120,  what  did  each  lose? 


6  INTRODUCTION  [Chap.  I. 

27.  The  greater  of  two  numbers  is  4  times  the  less.  Their 
sum  is  240.     What  are  the  numbers? 

Hint:   Let  n  equal  the  smaller  number. 

28.  The  sum  of  two  numbers  is  264.  The  greater  is  11 
times  the  less.     What  are  the  numbers? 

29.  The  sum  of  three  numbers  is  54.  The  second  is  twice 
the  first,  and  the  third  three  times  the  second.  Find  each 
number. 

30.  The  perimeter  of  a  rectangle  is  216  feet.  The  rectangle 
is  twice  as  long  as  it  is  wide.     What  are  its  dimensions? 

31.  A  man  buys  4  books  costing  x  cents  each,  and  2  balls  at 
X  cents  each.     The  whole  cost  one  dollar  and  a  half.     What  is  x? 

5.  Use  of  letters  to  abbreviate  statements.  As  pointed 
out  in  Art.  3,  the  use  of  letters  to  represent  numbers  enables 
us  to  write  many  statements  in  very  brief  form.  A  noticeable 
example  is  the  use  of  letters  to  abbreviate  the  rules  of  arith- 
metic. In  Art.  3,  the  rule:  —  The  area  of  a  rectangle  is  equal 
to  the  product  of  the  base  and  the  altitude  —  is  stated  in  the 
algebraic  form  A  =  h 

in  which  A  represents  the  area,  a  the  altitude,  and  b  the  base 
of  the  rectangle. 

Similarly,  if  we  were  given  the  length  I,  the  width  iv,  and 
the  height  h,  of  a  rectangular  solid,  the  rule  for  finding  the 
volume  is  expressed  in  the  form 
V  =  Iwh. 

The  rule  in  arithmetic  for  the  product  of  two  fractions  is: 
The  product  of  two  fractions  is  a  fraction  having  for  its  numera- 
tor the  product  of  the  two  numerators,  and  for  its  denominator 

the  product  of  the  two  denominators.     If  we  let  r  and  -,  repre- 
sent the  two  fractions  this  rule  becomes 

a     c__ac 

b'  d~Fd 


Arts.  6,  7,  S]  AREA   OF   A   TRIANGLE  7 

6.  Area  of  a  triangle.  In  arithmetic  it  was  learned  that 
the  area  of  a  triangle  is  equal  to  half  the  product  of  the  base 
and  altitude.  In  symbols 
bh 

"2' 


A 


where  A  is  the  area,  h  the  base, 
and  h  the  altitude. 

7.  Circumference  of  a 
circle.  In  arithmetic  it  was 
learned  that  the  circumference 
of  a  circle  equals  nearly  3.1416  times  the  diameter.  The 
number  3.1416  is  approximately  the  ratio  of  the  circumfer- 
ence to  the  diameter  and  is  denoted  by  the  Greek  letter  tt 
(pronounced  pi).     In  symbols 

c  =  rd, 

where  d  is  the  diameter  and  c  the  circum- 
ference. 

8.  Area  of  a  circle.     The  area  of  a  circle 
is  one  half  the  product  of  the  circumference 
Fig.  2  ^^^  radius.     Thus, 


A  = 


2' 


where  c  is  the  circumference,  r  the  radius,  and  A  the  area. 


EXERCISES  AND   PROBLEMS 

1.  If  A  is  the  area,  x  the  base,  and  y  the  altitude  of 
a  triangle,  write  in  algebraic  symbols  the  rule  for  finding  the 
area. 

2.  The  base  of  a  triangle  is  40  inches  and  the  altitude  is 
50  inches.     Find  the  area. 

3.  A  room  is  18  feet  long,  14  feet  wide,  and  8  feet  high. 
How  many  cubic  feet  of  air  are  there  in  the  room? 


8  INTRODUCTION  [Chap.  I. 

4.  State  in  algebraic  symbols  the  rule  for  the  length  c 
of  the  circumference  of  a  circle  when  the  radius  r  is 
given. 

5.  State  the  rule  for  the  area  of  a  circle  when  the  radius 
r  and  the  circumference  c  are  given. 

6.  What  is  the  interest  on  $300  for  one  year  at  the  rate 
of  6  %  per  annum. 

7.  Write  in  algebraic  symbols  the  rule  for  finding  the 
interest  on  a  sum  of  money  for  one  year.  Let  /  represent  the 
interest,  P  the  sum  of  money  in  dollars,  and  r  the  interest  on  a 
dollar  for  one  year. 

8.  What  is  the  simple  interest  on  $300  for  4  years  at  6  % 
annum? 

9.  If  P  is  the  principal,  r  the  rate,  and  n  the  number  of 
years,  write  in  letters  the  rule  for  finding  the  simple  in- 
terest. 

10.  State  in  words  the  rule  of  arithmetic  for  finding  the 

quotient  of  two  fractions.     Translate  this  rule  into  algebraic 

(l  .    .  c 

symbols  using  -r  as  the  dividend  and  ~j  as  the  divisor. 

11.  If  we  know  the  sum  of  two  numbers  and  one  of  the 
numbers,  how  may  we  find  the  other?  State  the  rule  in  alge- 
braic symbols.  Let  s  be  the  sum,  a  the  known  number,  and  x 
the  unknown  number. 

12.  State  in  algebraic  symbols  the  rule  for  finding  one  of 
two  numbers  when  their  product  and  one  of  the  numbers  are 
given. 

13.  Write  the  rule  for  finding  the  altitude  a  of  a  rectangle 
when  its  area  A  and  its  base  b  are  given. 

14.  The  volume  of  a  brick  is  64  cubic  inches.  Its  length  is 
8  inches,  its  width  4  inches.     What  is  its  thickness? 

15.  Write  the  rule  for  finding  the  thickness  t  of  a  brick, 
when  its  volume  V,  its  length  I,  and  its  width  w,  are  given. 

16.  Write  the  rule  for  finding  the  dividend  D,  when  the 
divisor  d,  and  quotient  q,  are  known. 


Arts.  0.  10,  u]  FACTORS  9 

9.  Factors.  Eacli  of  two  or  more  iiuiiibers  whose  product 
is  a  given  number  is  called  a  factor  of  the  given  number. 

Thus,  in  2  •  3  =  6,  2  and  .3  are  factors  of  6  ;  in  2  •  3  •  5  =  30, 
2,  3,  5,  G,  10,  and  15  are  factors  of  30.  Similarly,  5,  x,  y,  5x, 
by,  and  xy  are  factors  of  bxy. 

10.  CoeflEicient.  If  a  number  is  a  product  of  two  factors, 
then  either  of  these  two  is  called  the  coefficient  of  the  other  in 
the  product. 

Thus,  in  2  •  3,  2  is  the  coefficient  of  3,  and  3  is  the  coefficient 
of  2.  In  3a.T,  3  is  the  coefficient  of  ax,  3a  of  x,  x  of  3a,  and  a 
of  3.r. 

The  numerical  coefficient  1  is  usually  omitted.     Thus, 

1  •  a  =  a. 
In  such  expressions  as  'inx,  the  factor  consisting  of  Arabic  figures  is 
often  called  the  coefficient. 

EXERCISES 

1.  Give  factors  of  12  ;  of  14  ;  of  21;   of  36  ;  of  120. 

2.  Write  six  factors  of  2ab. 

3.  Write  ten  factors  of  Zahc. 

4.  Write  twelve  factors  of  Qxyz. 

5.  What  is  the  coefficient  of  x  in  Q)xy  ;  of  a  in  \a. 

6.  Write  down  factors  of  the  following  and  give  the  coeffi- 
cient of  each  factor.     Iz  ;   6a  ;   bxy  ;   8a6  ;   axy  ;   ^xz  ;   2a6c  ; 

f  I2ahc. 

11.    Exponents    and    powers.     An   exponent  is    a    number 
written  at  the  right  of  and  slightly  a])ove  another  number 
called  the  base.     When  the  exponent  is  an  integer,  it  shows  how 
often  the  base  is  taken  as  a  factor. 
Thus, 

42  =  4  •  4,   53  =  5  •  5  •  5, 
0?  =  a-a-a,   x^  ■  y*  =  x  •  x  ■  y  ■  y  ■  y  ■  y. 

A  power  of  a  number  is  the  product  obtained  by  using  the  num- 
ber as  a  factor  one  or  more  times. 


10  INTRODUCTION  [Chap.  I. 

We  may  read  a^  as  "a  exponent  m"  or  the  "rnih  power 
of  a."  Since  16  =  4^,  16  is  "4  exponent  2,"  or  the  "second 
power  of  4."  The  number  2^  is  read  "2  exponent  5"  or  the 
"fifth  power  of  2." 

When  the  exponent  is  1,  it  is  usually  omitted.  Thus,  a^  =  a. 
When  the  exponent  is  2,  the  power  is  called  the  square  of  the 
base,  and  when  the  exponent  is  3,  the  power  is  called  the  cube 
of  the  base.  Further,  a^  is  usually  read  "a  square,"  and  a^  is 
usually  read  "a  cube." 

The  meaning  of  an  exponent  that  is  not  one  of  the  integers, 
1,  2,  3,  .  .  .  will  be  explained  later  in  the  course  in  algebra. 

EXERCISES 

Write  the  following  products  using  exponents.     Read  the 

answers  aloud. 

1.  2  ■  2  •  2  •  2.  7.  a- a- a. 

2.  3  •  3.  8.  S-a-  X-  X. 

3.  5  •  5  •  5.  Q.  X  ■  X  ■  X  ■  X. 

4.  27.  10.  X  ■  X  ■  X  ■  a  •  a  •  a. 

5.  3  •  2  •  2.  11.  81  ■  a  ■  a  ■  X  ■  X  ■  b  ■  b. 

6.  7  •  3  •  3  ■  3  •  3.  12.  125  ■  x  ■  x  ■  x  ■  x  ■  a  ■  a  ■  a. 

Write  the  following  without  exponents. 

13.  61  16.    72.  19.    ay.  22.   a\ 

14.  3^  17.    9\  20.    (.02)^  23.   xhf. 

15.  4\  18.    2^  21.    (1)3.  24.   a^foV. 

If  a  =  2,  6  =  3,  aiid  n  =  5,  find  the  numerical  value  of  the 
following. 

25.  a2.  28.   6".  31.   a^  34.   a'^n«. 

26.  b\  29.    a^b^.  32.    n".  35.   3^bhi\ 

27.  2'K  30.    6^  33.    2hi'\  36.    Qa'^¥  -  aW. 

37.  What  exponent  is  understood  when  none  is  expressed, 
as  in  a,  or  x,  or  ax? 


Art.  u]  problems  U 


MISCELLANEOUS   PROBLEMS 

1.  The  second  of  three  consecutive  even  integers  is  n. 
What  are  the  other  two  numbers? 

2.  How  many  square  inches  are  there  in  a  rectangle  x 
feet  wide  and  y  feet  long? 

3.  A  brick  whose  height  is  x  inches,  is  twice  as  wide  as  it 
is  high,  and  twice  as  long  as  it  is  wide.     What  is  its  volume? 

4.  I  take  twice  as  long  to  walk  from  my  house  to  the 
station  uphill  as  to  return  downhill.  If  it  takes  9  minutes  to 
walk  there  and  back,  how  long  did  I  take  each  way? 

In  the  following  the  letters  stand  for  unknown  numbers. 
Find  the  numbers  and  check  the  results. 

5.  x  +  3=  11.  10.  3rc=  0.15.* 

6.  U=  x+  10.  11.  2y=b  +  3. 

7.  8  +  2  =  5  +  y.  12.  4z=  ^. 

8.  2x=  21.  13.  7y+3y=  1. 

9.  n  +  n=7  -  3.6. 

14.  A  certain  number  is  multiplied  by  3.  Twice  the  result 
is  42.     What  was  the  original  number? 

15.  Find  a  number  such  that  if  twice  the  number  be  added 
to  three  times  the  number,  the  sum  is  100. 

■  16.   The  sum  of  two  numbers  is  60.     The  greater  is  3  times 
the  smaller.     What  are  the  numbers? 

17.  Four  times  A's  money  is  $3000  more  than  B's.  B  has 
$8400.     How  much  has  Al 

18.  A  man  made  a  will  leaving  $10,000  to  be  divided  among 
3  daughters  and  4  sons.  Each  daughter  was  to  receive  twice 
as  much  as  each  son.     What  did  each  son  and  daughter  receive? 

*  Some  computers  prefer  to  write  0.1.5  in  place  of  .1.5.  The  form  0.15 
places  emphasis  on  the  fact  that  the  integi-al  part  of  the  number  is  zero 
and  removes  the  decimal  point  from  a  place  in  front  of  the  number  to  a 
position  where  it  is  not  so  likely  to  be  careles.sly  omitted.  In  writing 
decimals  we  shall  sometimes  use  the  one  form,  sometimes  the  other. 


12  INTRODUCTION  [Chap.  I. 

19.  A  man  bought  a  number  of  baseballs,  some  at  75  cents 
each,  and  twice  as  many  at  $1.25  each.  He  paid  $19.50  for  the 
lot.     How  many  of  each  did  he  get? 

20.  A  boy  has  $2.75  in  dimes  and  quarters.  He  has  3  times 
as  many  dimes  as  quarters.     How  many  of  each  has  he? 

21.  Which  would  you  rather  have,  3a:  +  8y  dollars,  or 
5x  +  6y  dollars  (a)  if  x  =  1000,  and  y  =  800?  (h)  If  a;  =  500, 
y  =  500?     (c)    If  a;  =  600,  y  =  700? 

22.  Write  in  symbols  :  The  sum  of  the  squares  of  a  and  h 
divided  by  the  cube  of  c. 

23.  Write  the  powers  of  10  from  the  first  to  the  tenth 
power. 

24.  The  number  343  is  what  power  of  7?  64  is  what  power 
of  8?   Of  4?    Of  2? 

25.  Write  as  a  power  of  12  the  number  of  cubic  inches  in  a 
cubic  foot. 

26.  Write  ten  factors  of  8aV. 

27.  Write  in  symbols  the  weight  of  a  rectangular  tank  of 
water.  Let  a,  b,  c  be  the  dimensions  of  the  tank  in  feet,  and  w 
the  weight  of  a  cubic  foot  of  water. 

28.  Using  the  answer  to  Problem  27,  find  the  weight  of  a 
tank  of  water,  for  which  a  =  2,  6  =  3,  c  =  10,  and  w  =  62.5, 

29.  A  man  saves  x  dollars  the  first  year.  During  the  second 
year  he  saves  100  dollars  more  than  in  the  first  year.  In  the 
third  year  he  lost  half  of  his  savings  for  the  first  two  years. 
What  were  his  net  savings  for  the  three  years? 

30.  If  V  is  the  volume  of  a  cone,  b  the  area  of  its  base,  and 
h  the  height,  it  is  known  that 

V  =  \bh. 

Give  this  result  in  words. 

12.  Historical  note  on  symbols.  The  history  of  the  early  use  of 
mathematical  symbols  is  very  interesting,  and  shows  how  mathematical 
progress  was  retarded  on  account  of  defects  in  symbolism.  To  appre- 
ciate in  a  slight  way  some  of  these  defects,  we  may  well  think  of  doing  a 


Art.  12]  HISTORICAL  NOTE  ON  SYMBOLS  L3 

fairly  long  calculation  with  Roman  numerals  I,  V,  X,  L,  C,  D,  M.  The 
so-called  Arabic  notation  that  uses  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  9,  0  is  of 
Hindu  origin,  the  Arabs  having  obtained  it  from  the  Hindus.  The  great 
achievement  of  inventing  a  satisfactory  method  of  writing  numbers  was 
not  accomplished  in  a  short  time.  It  required  centuries  to  perfect  this 
elegant  system.  While  the  dates  of  the  advances  are  in  doubt,  it  is  prob- 
able that  the  system  was  complete  as  early  as  the  sixth  century  of  the 
Christian  era. 

The  basic  idea  in  the  system  is  that  of  assigning  a  place  value  to  a 
digit.  A  symbol  for  zero  is  necessary  to  the  application  of  this  notion; 
but  the  importance  of  a  symbol  for  zero  was  not  recognized  until  long 
after  symbols  were  being  used  for  other  numbers.  After  the  principle 
of  a  place  value  was  established,  several  sets  of  characters  were  used.  In 
one  of  the  systems,  the  circle  O  was  used  to  denote  one  and  the  dot  •  was 
used  to  denote  zero.  It  is  held  by  some  that  the  symbols  used  for  digits 
were  perhaps  the  first  letters  of  early  numerals,  and  that  letters  were 
thus  used  to  denote  numbers  in  the  earliest  attempt  at  a  notation. 

The  symbols  +,  -,  x,  •,  -^,  =  came  into  common  use  in  the  first 
half  of  the  seventeenth  century,  and  their  origins  are  accounted  for  in 
various  ways.  I"  or  exanipIeTit  is  thought  by  some  that  the  sign  +  came 
from  the  rapid  writing  of  the  letter  p  in  the  word  plus,  and  by  others  that 
it  originated  in  warehouses  in  the  marking  of  boxes  that  showed  over- 
weight. Thus,  if  a  box,  supposed  to  weigh  50  pounds,  weighed  55  pounds, 
it  was  marked  50  +  5.  Whatever  may  be  correct  about  the  origin  of 
these  symbols,  it  is  doubtless  true  that  previous  to  their  use,  the  words 
for  which  they  stand  were  generally  written  out,  and  it  is  dear  that  the 
symbols  add  greatly  to  the  brevity  and  elegance  of  our  statements  involv- 
ing numbers. 


CHAPTER  II 
ALGEBRAIC  EXPRESSIONS 

13.    Algebraic   expressions.     In  algebra,   an   expression  is 

a  symbol  or  combination  of  symbols  that  represents  a  number. 

Thus,  gt  and  x  -  2y  +  ^z  are  expressions.  If  gr  =  32.2  and 
t  =  10,  the  first  has  the  value  322.  li  x  =  7,  y  =  2,  and  z  ^  4, 
the  second  has  a  value  15.  For  different  values  of  the  letters, 
the  expressions  usually  represent  different  numbers. 

EXERCISES 


If  a  =  1,  6  =  2,  c  =  5 

,   X 

=  1,  and 

y  = 

1,  find  the  value 

of 

each  of  the  following  algebraic  expressions 

1. 

a  +  2b  +  3c. 

5. 

a  + 

2bx  +  Sy. 

2. 

Sab  +  be. 

6. 

abxy  +  ~ 

3. 

5abc  +  6a. 

7. 

ab- 

-  xy. 

4. 

2ab  +  Sac  +  46c. 

8. 

^ab  +  Ixy  +  bc- 

3a.'r. 

9. 

Find   the  value 

of 

3a  +  2a6 

-bc, 

when  a  =  1 

b  = 

2 

and  c  = 

=  3. 

10. 

Find  the  value 

of  3a  +  2ab  - 

-be, 

when  a  =  \, 

b  = 

3, 

and  c  = 

=  1. 

11. 

Find  the  value 

of  3a  +  2a6  - 

-be, 

when  a  =  7, 

6  = 

h 

and  c  = 

"  ¥• 

14.  Order  of  operations.  If,  to  find  the  value  of  the  expres- 
sion, 4  +  5  •  6,  we  perform  the  operations  from  left  to  right  as 
we  come  to  each  symbol,  we  obtain  a  result  54.  If  we  perform 
the  multiplication  first,  we  obtain  the  result  34.  This  simple 
example  shows  that  results  depend  upon  the  order  in  which 
operations  are  performed. 

14 


Arts.  14,  lo]  ORDER  OF  OPERATIONS  15 

When  nothing  is  said  to  the  contrary,  it  is  understood  that 
in  a  series  of  operations  involving  additions,  subtractions, 
multiphcations  and  divisions,  the  multiphcations  and  divisions 
are  to  be  done  in  the  order  from  left  to  right  before  any 
additions  or  subtractions.  Then  additions  and  subtractions 
are  to  be  performed  in  any  order. 

Thus, 

4  +  5  •  6  =  4  +  30  =  34, 
and 

25  -  2  •  6  -  4  -  2  +  17  =  25  -  12  -  2  +  17  =  28. 

EXERCISES 

Carry  out  the  indicated  operations. 


1. 

34-2-3  +  4-2. 

5.    16-4-2-3. 

2. 

56-^2-2•4+6^2. 

6.   22-2+2-3. 

3. 

18  ^  9  -  2. 

7.   45-5-4-2  +  4 

4. 

18  -^  4  •  3  -^  2. 

8.    16-4-2-4-2. 

9. 

xyz  -  iz+2y  -3z  for  x  -- 

=  8,7/ 

=  2,z=l. 

15.  Use  of  parentheses.  To  group  the  parts  of  an  alge- 
braic expression  together,  parentheses  (  )  are  ordinarily  used. 

Thus, 

Sx+5y+  (x  +  y) 
means 

Sx  +  5y  +  x+  y. 

When  one  pair  of  parentheses  occurs  within  another,  it  is  con- 
venient to  use  different  forms  as  follows  :  [  ]  called  brackets  ; 
{  }  called  braces  ;  and  called  a  vinculum.     These  signs 

'are  often  called  signs  of  aggregation,  but  all  have  the  same 
meaning  and  may  be  called  parentheses. 

In  the  simplification  of  expressions  involving  parentheses,  it 
is  best  to  simplify  first  the  expression  within  the  parentheses. 
To  illustrate,  the  removal  of  the  parentheses  from  8  +  (4  -  2) 
gives  8  +  2  or  10.  The  expression  2(5  +  3)  means  2  times  8  ; 
a{h  +  c)  means  the  product  of  the  number  a  and  the  number 


16  ALGEBRAIC    EXPRESSIONS  [Chap.  II. 

obtained  by  adding  together  b  and  c.  The  expression 
(a  +  6)  (c  +  d)  means  the  product  of  the  sum  of  a  and  h  and 
the  sum  of  c  and  d. 

If   in  {a  +  b){c+  d),   a  =  2,   6  =  3,  c  =  5,    d  =  6,  we   have 
(2  +  3)(5  +  6)  =  5-  11  =  55. 

EXERCISES 

Simplify  the  following  : 

1.  10 +(4 +2). 

2.  7 +(5 +  2). 

3.  c+  (d+2c). 
Solution:   c  +  {d  +  2c)  =  c  +  d  +  2c  =  d  +  3c. 


4. 

{a  +  h)+ic+h). 

5. 

a: +  3(5 -2). 

6. 

a:  +  5(7-4). 

7. 

(18-2)-  (6-2). 

8. 

(8-4)(16-6). 

9. 

(7  +  3)(7-3)^(5-3). 

10. 

(7  +  3)(7-3)-5-3. 

11. 

(13-6)(18-8  +  8-^4). 

12. 

88  ^  4  -  10. 

13. 

(3  +  2)x  +  5(4  +  2)x. 

14. 

2x+Sy  +  Sx+  4y. 

15. 

2a  +  b  +  c  +  a  +  2b  +  2c. 

16. 

2  [x(2+6-4)  +  2.T  +  3x]. 

17. 

2i(3+4)  (5+1)+ (2  +  3)! 

18. 

5  [(3+7)  +  3  (1  +  2  +  3)]. 

16.  Evaluation  of  expressions.  It  is  frequently  necessary 
to  find  the  numerical  value  of  an  expression  for  certain  values 
of  the  letters  involved.  This  process  is  called  the  evaluation 
of  the  expression.  Such  evaluation  will  be  used  later  to  test 
the  accuracy  of  the  results  of  algebraic  operations  and  to  check 
the  answers  to  certain  problems. 


1. 

3a\ 

2. 
3. 

4. 

(a  +  b){c+d). 
(a  +  b)(c-d). 
4a-'6. 

5. 
6. 

2ab{c-d). 
Qab-b^  c. 

7. 

{a  +  by  -  2{c  - 

Arts.  16,  17]     EXPRESSIONS    IN   ONE    LETTER  17 


EXERCISES 

In  each  of  the  following,  find  the  value  of  the  expression 
for  a  =  3,  6  =  4,  c  =  5,  d  =  2,  and  simplify  the  results: 

9.  ab  +  bc  -  ad. 

10.  4«6  -  cd-  d. 

11.  6^  -  c  +  d-. 

12.  3c2  +  2a2  -  (/. 

13.  26  +  (5c-rf)-(a  +  6). 

14.  66  +  106c  ^  8a  -  f/. 
f/)2.                15.  4a62  _  cd2. 

8.   5c  ^  (6  -  d)  +  2(a  +  6).  16.   56c2  -  16(5a  -  36). 

17.  If  Q  represents  the  number  of  gallons  of  water  flowing 
from  a  pipe  per  minute,  v  the  rate  of  flow  of  the  water  in  feet 
per  minute,  and  d  the  diameter  of  the  pipe  in  inches,  it  is  known 
that 

Q  =  Mvd^. 

Find  the  number  of  gallons  of  water  flowing  per  minute  through 
a  pipe  2  inches  in  diameter  when  the  rate  of  flow  of  the  water 
is  1 00  feet  per  minute  ;  through  a  three-inch  pipe  when  the  rate 
is  75. 

17.  Expressions  containing  one  letter.  Algebraic  expres- 
''sions  which  contain  only  one  letter  form  a  class  of  great  im- 
portance. It  is  desirable  to  note  the  change  in  the  numerical 
value  of  the  expression  as  a  succession  of  numbers  is  substituted 
for  the  letter. 

Example.  Show  that  the  expression,  x^  +  12  -  7x,  decreases  as  x 
takes  on  the  succession  of  values,  0,  2,  1,  f,  2,  |,  and  3. 

Solution:  If  x  =  0,  then  x-  +  12  -  7x  equals  12. 

If  X  =  §,  then  X-  +  12  -  7x  equals  8^,  and  so  on. 

We  can  show  the  change  in  the  exi)ression  Ijy  a  tabic  as  f(jllo\vs: 


18 


ALGEBRAIC    EXPRESSIONS 


[Chap.   II. 


X 

x2  +  12  -  7x 

0 

12 

h 

8| 

1 

6 

f 

3i 

2 

2 

1 

1 

3 

0 

From  this  table  we  see  that  as  x  increases  from  0  to  3,  the  expression, 
x^  +  12  -  Tx,  decreases  from  12  to  0. 


EXERCISES 

1.  Find  the  value  of  2a:  +  3  f or  x  =  0  ;  f or  .t  =  1 ;  for  x  =  2  ; 
for  X  =  3. 

2.  Make  a  table  showing  the  values  of  5a;  -  3  when  x  takes 
on  the  values,  I,  2,  3,  4,  5. 

3.  Make  a  table  showing  the  values  of  7  -  3^  for  tj  =  0, 
.2,  .4,  .6,  .8,  and  1. 

4.  Show  that  52—1  increases  as  z  takes  on  the  succession 
of  values,  1,  f,  2,  |,  3,  f,  and  4. 

5.  Show  that  16  -  2^  decreases  as  z  takes  on  the  same 
values  as  in  Exercise  4. 

6.  Make  a  table  showing  the  values  of  x^  +  x  +  1,  for 
x  =  0,  1,  2,  3,  4. 

7.  Make  a  table  showing  the  values  of  x-  +  2  -  2x,  for 
a;  =  1,  2,  3,  4,  5,  6,  7.  Does  the  expression  increase  or 
decrease? 

8.  Make  a  table  showing  the  values  of  x^  +  2  -  2x,  for 
^  =  0>  4>  2j  f)  1-  Does  the  expression  increase  or  decrease  for 
these  values  of  x? 

9.  If  a  is  the  length  of  one  edge  of  a  cube,  and  V  is  the 
volume  of  the  cube,  write  in  symbols  the  rule  for  finding  the 
volume.  From  the  result  make  a  table  showing  the  volumes  of 
cubes  with  edges  ^,  1,  1|,  2,  2|  and  so  on  to  5. 

10.   If  $100  is  put  at  5  %  simple  interest  for  a  number  of 
years  it  amounts  to  100  +  5/;  dollars  if  n  is  the  numlier  of  years. 


Arts.  17,  18]     GRAPHICAL  REPRESENTATION 


19 


From  this  make  a  table  showing  how  $100  increases  every  year 
up  to  10  years. 

11.  Show  in  symbols  how  $100  increases  if  put  at  6  % 
simple  interest.     Tabulate  the  results  as  in  Problem  10. 

12.  If  y  represents  the  velocity  of  the  wind,  and  P  is  the 
pressure  of  the  wind  on  a  pane  of  glass  one  foot  square,  then 
it  is  known  that 

P  =  v'  -  225. 

Find  the  pressure  when  the  wind  is  blowing  at  the  rate  of  15 
miles  per  hour  ;   20,  30,  40,  50  miles  per  hour. 

18.  Graphical  representation.  The  way  in  which  an  ex- 
pression involving  one  letter  changes  as  the  value  assigned  to 
the  letter  is  changed,  can  be  represented  to  the  eye  by  a  dia- 
gram. This  is  shown  as  follows  for  the  expression  2x  +  3.  On 
the  horizontal  line  of  Fig.  3  mark  the  numbers  1,  2,  3,  4  .  .  . 
Make  a  table  of  values  for  2a:  +  3.     Thus  we  have: 


X 

1 

2 

3 

4 

5 

6 

2x  +  3 

5 

7 

9 

11 

13 

15 

At  the  point  marked  1  draw  a  vertical  line  of  length  2x+3 
when  X  =  I;  that  is,  of  length  5.     At  the  point  marked  2  erect  a 


Fig.  3 


vertical  line  of  length  2x  +  3  when  x  =  2;  that  is,  of  length  7. 
Proceed  in  the  same  way  for  each  value  of  x  given  in  the  table. 


20 


ALGEBRAIC  EXPRESSIONS 


[Chap.  II. 


The  diagram  will  then  present  to  the  eye  the  way  in  which 
2x  +  3  increases  as  x  increases. 

Any  table  of  numbers,  whether  representing  an  algebraic 
expression  or  not  may  be  represented  to  the  eye  in  a  like  manner. 
For  example  :  During  the  opening  day  of  a  new  shoe  store  a 
record  was  kept  of  the  number  of  customers  for  each  hour  of 
the  ten  hours  the  store  was  open. 


Hour 

1|2|3|4|5|6|7|8|9|10 

Number  of  customers 

2|4|8|6|2|3|9|7|6|     6 

This  table  is  represented  in  Fig.  4. 

10- 
9 


i  6^- 

i   5- 

U 


4  5  6 

UuuTB  during  nhioh  the  Store  is  c 

Fig.  4 


In  these  diagrams  any  convenient  unit  can  be  used  for  the 
lengths  of  the  vertical  lines. 

EXERCISES 

1.  Show  by  a  diagram  the  values  of  5a;  -  3  when  x  takes  on 
the  integral  values  1  to  6. 

Represent  by  a  diagram  the  following  expressions  when  x 
takes  on  the  integral  values  1  to  6. 

2.  11  -  a;.  4:.   X-  +  X+  1. 

3.  3x-  1.  5.   a:2  +  2-2x. 

6.   The  morning  temperature  record  of  a  fever  patient  is 
given  by  the  table: 


Art.  18]  GRAPHICAL  REPRESENTATION  21 


Morning                          |1|2     |3|4|5     |6|7|8|9! 

Degrees  above  normal 
temperature 

1 

3.5 

'' 

2 

1.8 

1 

.9 

.6     0 

Represent  this  table  by  a  diagram. 

7.   The  weights  of  a  baby  weighing  eight  pounds  at  birth 
for  each  month  of  the  year  are  given  by  the  table: 


Month    I  1    I  2      I  3      I    4    I    5    i    6    I    7  I    8  I    9    I  10  I  11  i  12 


Weight  i  9i  I  lU  I  12f  I  14i  I  15^  |  16^  |  18  |  19  |  19^  |  20  |  21  |  22 


Represent  by  a  diagram. 

8.  The  heights  of  the  same  baby  for  the  same  months  are 
19  inches  at  birth,  then  20^  21,  22,  23,  23^,  24,  241,  25,  25|, 
26,  26^,  27  inches.     Represent  by  a  diagram. 

EXERCISES  AND   PROBLEMS 

1.  The  horse  power  of  a  certain  style  of  gasoline  engine 
is  given  by  the  expression 

H  =  ID^~N* 
where  H  is  horse  power,  D  the  diameter  of  the  cylinder  in  inches, 
and  N  the  number  of  cylinders.     Find  the  horse  power  of  a  two- 
cylinder  engine  with  5-inch  cylinders;  with  6-inch  cylinders. 

2.  Write  in  algebraic  symbols  :  The  volume  of  a  sphere  is 
four  thirds  the  cube  of  the  radius  times  tt. 

3.  5  •  22  -  33  -=-  3  =  ?  4.   22  •  3'^  -  18  •  52  -  32  =  ? 

5.  In  the  formula  s  =  16.1^2^  i  represents  the  number  of 
seconds  a  body  has  been  falling,  and  s  represents  the  distance 
it  has  fallen.     How  far  will  a  stone  fall  in  4  seconds? 

6.  Give  the  formula  of  Problem  5  in  words. 

7.  Make  a  table  showing  the  distance  through  which  a 
stone  will  fall  in  1,  2,  3,  4,  5,  6,  7,  8,  9,  10  seconds. 

Find  the  values  of  the  letters  in  the  following: 

8.  5x  =  25.  10.   32  =  84. 

9.  4  +  rt=  19.  11.   3x-h2a:=20. 


*  It  is  not  expected  in  this  and  similar  problems  that  the  teacher  will 
take  the  time  to  explain  the  physical  principles  involved. 


22  ALGEBRAIC    EXPRESSIONS  [Chap.  II. 

12.  A  has  $550  and  B  $150.  How  much  must  A  give  to 
B  in  order  to  have  just  twice  as  much  as  B? 

13.  If  a  is  equal  to  2  times  b,  express  6a  +  36  in  terms  of  b. 

14.  Write  6  factors  of  2x{a  +  b). 

15.  4a;2  -  3x^  +  5x^  -  x^  =  ? 

16.  If  a;  is  3  greater  than  y,  express  7x  +  2y  first  in  terms 
of  x  and  then  in  terms  of  y. 

17.  Find  the  value  of  x{[{x  +  l)- {y  -  \)] -\- xy\  when 
X  =  2  and  ^  =  4. 

18.  7  (a  -  6)  -  2(a  -  b)  -  3(a  -  6)  =  (?)  •  (a  -  6). 

19.  Write  in  symbols  :  The  square  of  the  sum  of  a  and  b 
divided  by  their  product. 

20.  Find  an  expression  for  the  surface  of  a  cube  if  one  edge 
is  given.  From  the  result  make  a  table  showing  the  surfaces 
of  cubes  with  edges  1,  2,  ...  to  6. 

21.  A  bag  contains  x  white  balls,  three  times  as  many  black 
balls  as  white  balls,  and  twice  as  many  red  balls  as  black  balls. 
How  many  are  there  of  each  color  if  there  are  40  balls  altogether? 

22.  Given  that  254  centimeters  equals  100  inches  very 
nearly.  Using  c  for  number  of  centimeters  and  i  for  number 
of  inches,  show  in  symbols  the  relation  between  inches  and 
centimeters.     Tabulate  the  result  up  to  12  inches. 

23.  Show  by  a  diagram  the  temperature  record  of  a  fever 
patient.  The  degrees  of  fever  recorded  were  2,  5.4,  4,  6.1,  4.5, 
6.5,  5.3,  6.7,  4.5,  6.5,  5.9,  6.2,  7,  5,  6.7,  5.3,  6.1,  7.4. 

24.  The  following  table  gives  the  length  of  a  child's  bare 
foot  when  the  size  of  the  shoe  is  known. 


Size  of  shoe |  1    |2|  3    |  4    |  5|  6    |  7    |  8  |9    |10|11  |12|13 

Length  of  foot  in  inches  |  3|  |  4  |  4^  |  4f  |  5  |  5^  1  51  |  6  |  Gj  |  6f  |    7  |  7^  1  71 


Show  by  a  diagram  the  relation  between  size  of  shoe  and  length 
of  foot  for  children. 


CHAPTER    III 
POSITIVE   AND   NEGATIVE   NUMBERS 

19.  The  use  of  a  scale  to  represent  the  numbers  of  arith- 
metic. The  numbers  of  arithmetic  may  be  represented  by 
points  on  a  straight  line. 

Let  OX  be  such  a  line  (Fig,  5).  Adopt  a  unit  of  measure, 
AB.  Begin  at  0  and  divide  OX  into  intervals  of  length  AB. 
We  may  thus  obtain  a  scale  of  indefinite  length.  If  the  ends 
of  the  intervals  are  marked  0,  1,  2,  3,  4,  ...  as  in  the  figure, 

J L     ol \ \ 1— J I \ I ! 1 X 

AB012345G789 

Fig.  5 

we  have  a  point  on  the  number  scale  corresponding  to  each  of 
these  numbers. 

Fractions  may  also  be  represented  by  points  on  this  line. 

Thus,  there  corresponds  to  the  number  |  a  point  midway 

between  the  points  marked  0  and  1  ;  to  3^  a  point  ^  of  the  dis- 

^  tance  from  3  to  4  ;  and,  in  the  same  way,  for  every  fraction 

there  corresponds  a  point  on  the  scale. 

20.  Addition  and  subtraction  on  the  scale.  Addition  and 
subtraction  may  be  performed  on  the  number  scale. 

Thus,  to  add  3  to  4,  begin  at  4  and  count  3  spaces  to  the 
right.  To  subtract  3  from  4,  begin  at  4  and  count  3  spaces  to 
the  left. 

In  general,  to  add  a  to  h,  begin  at  b  and  count  a  spaces  to 
the  right  ;  and  to  subtract  a  from  b,  begin  at  b  and  count  a 
spaces  to  the  left. 

23 


24  POSITIVE   AND    NEGATIVE   NUMBERS     [Chap.  III. 

EXERCISES 

1.  What  number  is  represented  by  each  of  the  following 
points? 

(a)  The  point  3  spaces  to  the  right  of  5. 
(6)  The  point  4  spaces  to  the  left  of  II. 

(c)  The  point  midway  between  4  and  5. 

(d)  The  point  midway  between  3^  and  4. 

(e)  The  point  one-third  of  the  distance  from  2  to  3. 

2.  State  in  words  the  positions  of  the  points  that  represent 
the  numbers  |,  I,  |,  J,  V-,  0. 

3.  On  an  ordinary  ruler  perform  the  following  additions 
and   subtractions  : 

(a)  5+2.  (e)   2+3-4.  (h)  21+1. 

(6)  5-2.  (/)    7-2-3.  (^•)     8-2-i. 

(c)   5-5.  (g)   i+i-  (j)     6-4-2. 

id)  2  +  3  +  4. 

21.  Positive  and  negative  numbers.  If,  on  the  scale  in 
Fig.  5,  we  attempt  to  subtract  a  number  from  a  smaller 
number,  say  6  from  4,  we  get  off  the  scale. 

Let  the  scale  be  extended  to  the  left  of  0,  Fig.  6.  If  we  now 
attempt  to  subtract  6  from  4  by  counting  6  units  to  the  left  of 
4,  we  arrive  at  a  point  2  units  to  the  left  of  0. 

I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    '    I    I    I 

-11-10-9-8-7-6-5-4-3-2-1      0      1     2     3     4     5     6     7     8     9     10    11 

Fig.  6 
The  numbers  that  correspond  to  the  points  to  the  left  of  0 
are  called  negative  numbers. 

The  numbers  of  arithmetic,  which  are  represented  by  points 
to  the  right  of  0,  are  called  positive  numbers. 

The  minus  sign  is  used  to  distinguish  the  negative  from  the 
positive  numbers. 
On  this  scale  then, 

4-6=  -2. 


Akts.  21,22,23,24]   POSITR-E  AND  NEGATIVE  NUMBERS    25 

Similarly, 

7  -  10  =  -3,  and  2  -  8  =  -G. 

The  essential  thing  in  the  above  representation  of  positive 
and  negative  numbers  is  that  they  are  marked  off  in  opposite 
directions. 

In  Art.  1,  we  ol^serve  that  the  fraction  is  introduced  to  make 
division  possible  when  the  quotient  is  not  an  integer.  In  a 
similar  manner,  the  negative  number  is  here  introduced  to  make 
it  possible  to  subtract  from  a  number  a  greater  number. 

22.  Illustrations.  The  student  has  had  experience  in  arith- 
metic with  quantities  measured  in  opposite  directions. 

The  thermometer  gives  a  simple  example.  Temperatures 
above  and  below  zero  are  measured  in  opposite  directions.  It 
is  simple  and  convenient  to  regard  temperatures  above  zero  as 
positive  and  those  below  zero  as  negative. 

If  the  temperature  is  10°  above  zero,  or  +10°,  and  falls 
30°,  we  say  it  is  then  20°  below  zero,  or  -20°.  This  result  is 
obtained  algebraically  by  saying  10°  -  30°  =  -20°. 

North  latitude  is  commonly  called  positive  and  south 
latitude  negative.  If  a  ship  is  in  latitude  +20°  and  sails  30° 
south,  it  is  then  in  latitude   -10°.     Algebraically, 

20°-  30°=    -10°. 

Positive  and  negative  numbers  are  often  called  signed 
numbers. 

23.  Numerical  or  absolute  value.  The  numerical  or  ab- 
solute value  of  a  number  is  its  value  without  regard  to  sign. 

The  absolute  values  of  -10,  +30,  +6,  -9,  are  10,  30, 
6,  and  9,  respectively. 

24.  Greater  and  less.  The  expressions  greater  than  and 
less  than  which  are  common  to  every  day  life,  when  used  in  the 
precise  sense  of  algebra,  are  easily  misunderstood.  For  this 
reason,  we  point  out  their  meanings  on  the  number  scale.    A 


26  POSITIVE    AND    NEGATIVE    NUMBERS     [Chap.  Ill 

number  a  is  greater  than  a  number  h  when  a  is  represented  by 
a  point  on  the  scale  to  the  right  of  that  representing  h. 
Thus,  1  is  greater  than  -4  ;  a  temperature  of  -10°  is  greater 
(or  higher)  than  one  of  -20°. 

\ \ \ \ \ \ \ \ \ I    •     ' 

-5-4-3-2-1  0  1  2  3  -i  3 

Fig.  7 

Exercise.     Explain  by  the  use  of  points  on  the  number  scale  what  is 
meant  by  the  expression  "a  is  less  than  6." 

EXERCISES   AND   PROBLEMS 

1.  Locate  on  the  number  scale:  —5,  -2,  0,  -f,  -f, 
+5. 

2.  Find  the  point  that  corresponds  to  the  following  dif- 
ferences :    5-11;   7-7;    4-4.5;    6-5;    0-4. 

3.  Using  temperature  above  zero,  north  latitude,  west 
longitude,  and  assets  as  positive  quantities,  represent  the 
following  as  signed  numbers  :  12°  above  zero  ;  $10  debts  ; 
30°  west  longitude  ;  40°  south  latitude  ;  $100  assets  ;  1°  below 
zero. 

4.  A  goes  6  miles  west,  then  18  miles  east,  then  5  miles 
west,  then  2  miles  east.  Using  distance  east  as  positive,  write 
the  above  distances  as  signed  numbers.  How  far  and  in  what 
direction  is  he  from  the  starting  point?  Show  how  to  find  the 
answer  by  using  the  number  scale. 

5.  If  the  temperature  is  +10°,  falls  13°,  and  then  rises 
5°,  what  is  the  resulting  temperature? 

6.  The  temperature  is  now  60°.  What  will  it  be  after  it 
falls  :   (a)  20°  ;   (6)  60°  ;   (c)  70°? 

7.  If  the  temperature  is  - 10°,  what  will  it  be  after  it  rises : 
(a)  5°  ;   (6)  10°  ;   (c)  20°? 

8.  Give  the  absolute  values  of  +8,  -10,  -^,  +f, 
-8.5. 

9.  What  number  is  one  greater  than  -8?  What  number 
is  one  less? 


Art.  24]  EXERCISES   AND    PROBLEMS  27 

10.  Show  on  the  number  scale  the  distance  between  (a) 
+7  and  +4  ;  (6)  -7  and  -4  ;  (c)  -7  and  +4  ;  (d)  +7  and  -4. 

11.  In  a  game,  A  loses  10  points,  gains  7,  gains  31,  loses  14, 
loses  36,  gains  5.     What  is  his  final  score? 

12.  A  has  $375  assets  and  no  debts  ;  B  has  $200  debts  and 
no  assets.  Consider  debts  as  negative  assets.  What  is  the 
sum  of  their  assets?     How  much  more  assets  has  A  than  5? 

13.  What  should  be  taken  negative  if  the  following  are  con- 
sidered  positive? 

(a)  West  longitude.  (/)  Cubic  inches  of  expan- 

(b)  Dollars  gain.  sion  of  a  toy  l^allon. 

(c)  Points  by  which  a  game       (g)  Excess  of  water  pumped 

is  won.  from  a  well  over  that 

(d)  Miles  northeast.  flowing  into  it  in  the 

(e)  Excess  of  persons  enter-  same  time. 

ing  a  room  over  those 
leaving  the  room. 

14.  What  would  be  the  meaning,  if  any,  of  a  negative  result 
in  finding  — 

(a)  How  much  money  you  had  gained  in  a  trade? 
(6)  The  score  by  which  your  football  team  won  the  last 
game  of  the  season? 

(c)  How  much  more  a  cubic  foot  of  a  certain  liquid  weighs 

than  a  cubic  foot  of  water? 

(d)  The  rate  at  which  a  lion  ran  away  from  a  hunter? 

(e)  The  increase  of  the  population  of  a  town  in  10  years? 
(/)    How  far  Philadelphia  is  west  of  New  York? 

Historical  note  on  negative  numbers.  The  extension  of  the  concep- 
tion of  number  to  inchide  the  negative  was  an  excooclingly  slow  process 
in  the  development  of  algebra.  The  Himlu.s  recognized  the  exist- 
ence of  the  negative  in  working  at  the  quadratic  equation  perhaps  as  early 
as  800  A.D.,  but  they  had  httle  to  say  about  such  numbers  except  that 
the  people  did  not  approve  of  them.  It  was  not  until  the  work  of  Des- 
cartes (1596-1G50)  (see  p.  182)  that  the  rules  for  operation  with  negative 
numbers  were  at  all  well  understood. 


28  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 

(g)  The  height  of  the  bottom  of  a  well  above  sea-level? 
(h)  The  age  of  Washington  at  the  time  of  the  surrender 
of  Yorktown? 

25.  Addition  of  signed  numbers.  Thus  far  it  has  not  been 
indicated  what  is  meant  by  the  sum  of  two  numbers  when  one  or 
both  are  negative.  Some  illustrations  will  suggest  how  such 
sums  should  be  defined. 

Example.  How  many  steps  and  in  what  direction  from  the  starting 
point  is  a  person  who  takes  — 

(a)  8  steps  east  and  then  5  steps  east? 
(6)  8  steps  west  and  then  5  steps  west? 

(c)  8  steps  east  and  then  5  steps  west? 

(d)  8  steps  west  and  then  5  steps  east? 

Solution:  Let  steps  east  be  considered  positive  and  steps  west  nega- 
tive.    The  answers  may  be  obtained  by  counting  on  a  scale. 

(a)  To  add  +5  to  +8  we  begin  at  +8  and  count  5  units  to  the  right, 
arriving  at   +13. 

(6)  To  add  -5  to  -8  we  begin  at  -8  and  count  5  units  to  the  left, 
arriving  at   -13. 

(c)  To  add  -5  to  +8  we  begin  at  +8  and  count  5  units  to  the  left, 
arriving  at   +3. 

(d)  To  add  +5  to  -8  we  begin  at  -8  and  count  5  units  to  the  right, 
arriving  at   -3. 

Hence  to  add  a  positive  number  we  count  to  the  right,  mid  to  add  a  nega- 
tive number  we  count  to  the  left. 

The  results  just  obtained  may  be  stated  as  follows: 

+8  +  (  +5)  =   +13, 

-8  +  (  -5)  =   -13, 

+8  +  (  -5)  =   +3, 

-8  +  (  +5)  =    -3. 

The  foregoing  discussion  suggests  the  following  definitions 
of  sum  when  signed  numbers  are  added  : 

The  simi  of  numbers  with  like  signs  is  the  swu  of  their  absolute 
values  preceded  by  their  common  sign. 

The  sum  of  numbers  with  unlike  signs  is  the  difference  of 
their  absolute  values  preceded  by  the  sign  of  the  one  having  the 
greater  absolute  value. 


Arts.  25,  2G]      ADDITION  OF  SIGNED  NUMBERS  29 

EXERCISES 
Perform  the  following  additions  by  counting  on  a  scale: 

1.       10           5.       6               9.  -7           13.      3  15.         8 

J                 ^                   J2                   -4  -9 

2  4 

10.  -6                      1  -2 
6 

11.  -8            14.      9  16.       10 
-8                  -6  -12 

-3  4 

12.  -9  _jl  6 

4 


22.  -4=  -13+? 

23.  ?  +  6  =  0. 

24.  -12+?=0. 

25.  3=  -7+  ? 


Find  the  value  of  x  in  the  following  : 

26.  a;+    8=  12.  31.     -1  -  x  =  -3. 

27.  a:  -    7  =    3.  32.     20  +  a:  =  0. 

28.  a; -11  =  -8.  33.     13  -  a;  =  0. 

29.  4-a:=-16.  34.     .r+(-ll)=15. 

30.  15  =  .T-  9.  35.      .T  -2  -  -2. 

26.  Subtraction  of  signed  numbers.  As  in  arithmetic, 
subtraction  is  the  process  of  finding  one  of  two  numbers  when 
their  sum  and  the  other  number  are  given. 

Thus,     +10  -  (+ 4)  =  6  because        6  +  4         =  10. 

-10-  (  +  4)=  -14  because   -14  +  4         =  -10. 

+10  -  (  -4)  =  14  because      14  +  (  -4)  =  10. 

-10  -  (  -4)  =  -  6  because     -6  +  (  -4)  =  -10. 


10 

6.       5 

-7 

-8 

10 

7.    -2 

7 

-12 

10 

8.       4 

-7 

-5 

nswer  the  following 

17. 

8  +  ?  =  13. 

18. 

?  +  8  =  6. 

19. 

?+(-5)  =  7 

20. 

-8+?=  12. 

21. 

?+5=   -10. 

30  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 


EXERCISES 

Fill  in  the  blanks  in  the  following  : 

1.  +7  -  (  +4)  =  (  ),  since  ( +4)  +  (  )  =  +7. 

2.  -7  -  (  -4)  =  (  ),  since  (  -4)  +  (  )  =  -7. 

3.  +4  -  (  +7)  =  (  ),  since  (  +7)  +  (  )  =  +4. 

4.  +7  -  (  -4)  =  (  ),  since  (  -4)  +  (  )  =  +7. 

In  a  similar  Avay  perform  the  following  subtractions 


5. 

3- 

-(+2). 

6. 

-3- 

-(+2). 

7. 

-3- 

-(-2). 

8. 

5- 

-(-5). 

9. 

-4- 

-(-4). 

10. 

-9- 

-  (10). 

11. 

-1  - 

(+2). 

12. 

-1- 

(-!)• 

13. 

8- 

-(+8). 

14. 

7- 

-(+9). 

15. 

0- 

-(-7). 

16. 

0- 

-(+7). 

17. 

27- 

-(-9). 

18. 

-8- 

-(+18) 

19. 

30- 

-(-8). 

20.     -8-  (+30). 

Answer  the  following  questions  : 

21.  12-?=0.  25.         8-    ?=-5. 

22.  ?  -  6  =  0.  26.       -7  -    ?  =  4. 

23.  -12-?  =  0.  27.       -5-14=? 

24.  -8-?=  4.  28.    -10-    ?=5. 

21.  Subtraction  on  a  scale.  We  have  seen,  Art.  20,  that 
to  subtract  a  positive  number  on  the  scale  we  count  to  the  left. 
Thus,  in  Art.  26,  the  result  in  Exercise  1  can  be  found  by 
beginning  at  +7  and  counting  4  spaces  to  the  left,  arriving  at 

+3  ;   and  the  result  in  Exercise  3  can  be  found  by  beginning 
at  +4  and  counting  7  spaces  to  the  left,  arriving  at  -3. 

We  can  also  subtract  a  negative  number  by  counting  on  the 
scale.  The  answer  to  Exercise  2,  Art.  26,  can  be  found  by 
beginning  at  -7  and  counting  4  spaces  to  the  right,  arriving  at 

-3  ;   and  the  result  in  Exercise  4  can  be  found  by  beginning 
at  +7  and  counting  4  spaces  to  the  right,  arriving  at  +11. 


Art.  2S] 


RULE  FOR  SUBTRACTION 


31 


We  have  then  the  i)rinciple :  To  subtract  a  negative 
number,  count  to  the  right  the  number  of  spaces  indicated  by 
the  number. 

Exercise.  Find  the  answers  to  Exercises  5-20  in  Art.  26  by  counting 
on  the  scale. 


28.    Rule    for    subtraction.     Any    problem  in  subtraction 
may  be  changed  to  a  problem  in  addition. 


Examples. 


+10  -  (  +5)  =  +10  +  (  -5)  =   +5. 

-10  -  (  -5)  =  -10  +  (  +5)  =   -  5. 

-10-  (+5)  =  -10 +(-5)  =   -15. 

+10  -  (  -5)  =  +10  +  (  +5)  =   +15. 


We  then  have  the  working  rule  :    To  subtract  amj  number 
change  its  sign  and  add  the  resuUing  number. 


EXERCISES 
Perform  the  following  additions  : 


1.    -11 


11 


3.    -11 


4. 


13 

7. 

-3 

9. 

-1 

-9 

2 
-5 

3 

-7 

9 

13 

IS 

8. 

-13 
-2 

10. 

-11 

7 

17 

6 

4 

19 

-3 

-21 

Perform  the  following  subtractions  : 

11.  3-7.  14.   9-17.  17.  -3    -(-14). 

12.  3 -(-11).        15.   9 -(-17).  18.  -11-18. 

13.  7-8.  16.   2  -  (-2).  19.  -|    -  (-i). 


20. 

13-?  =  1. 

27. 

21. 

13  -  ?  =  -1. 

28. 

22. 

1  -  13  =  ? 

29. 

23. 

9  -  3  +  ?  =  2. 

30. 

24. 

1-2+3+?= 

=  4. 

31. 

32  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 

Answer  the  following  questions  : 

-?-(-2)  -(-11)  =  -6. 
-2-(-i)  +  ?  =  f- 
-17  -  (-11)  +5  =  ? 
l-(-2)  +  (-3)-(-4)  +  ?  =  0. 
10  +  ?  -  16  =  -6. 

25.  -21  +  11  -  ?  =  14.  32.    ?  +  (-7)  -  (-7)  +  7  =  -7. 

26.  -3  -  ?  -  (-7)  =  0.  33.    -.1  +  .5  -  (-.6)  -  (?)  =  1. 

EXERCISES   AND   PROBLEMS 

1.  From  the  sum  of  -2|,  -3^,  and  12f,  subtract  the  sum 
of  3,  5,  and  8^. 

2.  What  is  the  difference  between  the  temperatures  of 
+85°  and  +40°?  Of  +36°  and  +14°?  Of  +43°  and  -20°? 
Of  -43°  and  -51°?     Of  -24°  and  -6°? 

3.  If  a  man  is  worth  $20,000,  how  much  must  he  lose  to 
be  in  debt  $8400? 

4.  B  has  assets  of  $300,  $100,  $800,  and  $1000,  and  debts 
of  $200,  $400,  $100,  and  $600.  Let  assets  be  represented  by 
positive  numbers  and  debts  by  negative  numbers.  What  is 
the  net  value  of  5's  property?  What  would  be  the  net  value 
of  his  property  if  (a)  the  courts  should  cancel  the  debts  of 
$300  and  $100;  (6)  he  should  lose  $400  worth  of  property  by 
fire;     (c)  he  should  gain  $400  in  a  trade? 

5.  Suppose  that  weights  to  the  amount  of  300  pounds  are 
attached  to  a  balloon  which  pulls  upward  with  a  force  of  400 
pounds.  Let  pull  upward  be  regarded  as  positive  and  pull 
downward  (weight)  as  negative.  What  is  the  net  upward 
pull?  What  would  be  the  net  upward  or  downward  pull 
if  (a)  200  pounds  of  weights  were  removed;  (6)  gns  with 
a  lifting  force  of  200  pounds  were  added;  (c)  200  pounds  of 
weights  were  added ;  {d)  gas  with  a  lifting  force  of  200  pounds 
were  removed? 

6.  In  the  difference  a  -  6  =  6,  the  value  of  a  runs  through 
the  series  of  whole  numbers  from  10  to  0.     Find  the  correspond- 


Art.  28]  EXERCISES  AND  PROBLEMS  33 

ing  values  of  b.     Make  a  table  showing  these  corresponding 
values  of  a  and  b. 

7.  In  the  difference  x  -  y  =  -4,  the  value  of  ij  runs 
through  the  series  of  whole  numbers  from  -5  to  +5.  Find  the 
corresponding  values  of  x,  and  make  a  table  as  in  Exercise  6. 

In  the  following  problems  +  denotes  A.D.,  and  -  denotes 
B.C. 

8.  The  oldest  mathematical  manuscript  known  was  writ- 
ten by  an  Egyptian  named  Ahmes.  The  date  of  the  manu- 
script is  thought  to  be  about  -1700.     How  old  is  it? 

9.  The  Greek  geometer,  Euclid,  lived  about  -300.  How 
long  was  that  before  the  birth  of  the  French  geometer,  Des- 
cartes, who  was  born  in  +1596? 

10.  Some  characters  from  which  our  present  numerals  are 
thought  to  have  developed,  are  found  in  inscriptions  made  in 
India  as  early  as  -250.  The  first  undoubted  use  of  the  zero 
in  India  is  said  to  have  been  in  the  year  +876.  How  many 
years  between  these  dates? 

11.  Archimedes  was  born  in  the  year  -287,  and  died  in 
the  year  -212.     How  old  was  he  when  he  died? 

12.  Through  how  many  degrees  of  latitude  does  a  ship  sail 
in  going  from  latitude  -21°  to  latitude  -56°? 

13.  Through  how  many  degrees  of  longitude  does  a  ship 
sail  in  going  from  longitude  —17°  to  longitude  +35°? 

?j        State  what  single  change  will  produce  the  same  result  as  the 
changes  mentioned  in  each  of  the  following  exercises. 

14.  The  temperature  rises  20°,  falls  13°,  then  rises  6°. 

15.  An  elevator  goes  up  60  feet,  goes  down  24  feet,  goes  up 
48  feet,  and  then  goes  down  84  feet. 

16.  A  traveler  goes  350  miles  east,  and  then  420  miles 
w^est. 

17.  In  a  race  a  runner  gains  12  feet  on  his  opponent,  loses 
7  feet,  gains  2  feet,  and  loses  9  feet. 

18.  A  political  party  gains  724  votes,  loses  328,  loses  35, 
and  gains  120. 


34  POSITIVE  AND  NEGATIVE  NUMBERS      [Chap.  III. 

39.  Addition  and  subtraction  of  several  numbers.  In  add- 
ing and  subtracting  several  numbers  we  may  proceed  from  left 
to  right  performing  each  addition  and  subtraction  as  we  come 
to  it.     For  example, 

4  +  3  +  (-2)  -  3  -  (-1)  =  7  +  (-2)  -  3-  (-1) 
=  5-3-  (-1) 
=  2-(-l) 
=  3. 
It  is  usually  shorter,  however,  to  remove  the  parentheses, 
unite  the  positive  terms  and  the  negative  terms,  and  then  com- 
bine these  results. 

Thus,     4  +  3  +  (-2)  -  3  -  (-1)  =4+3-2-3  +  1 

=  8-5 
=  3. 
The  second  method  is  usually  more  convenient  when  the 
numbers  are  written  in  columns. 

EXERCISES 
Perform   the  following  additions  and   subtractions : 

1.  2-3  +  4-5  +  6-7  +  8. 

2.  -2  -  (-3)  +  (-4)  -  (-5)  +  (-6)  -  (-7). 

3.  l  +  (-2)  -3  + (-4)  -  (-5). 


7.      - 


8. 


4.   2a- 

-3a 

+  a  -  4a  +  6a 

5.    5x  - 

-  Qx  +  X  -  7x  +  2x 

-  (-4x). 

6.    -1- 

-2- 

-3-(-4)-(- 

5)  -  (-6). 

-8 

9.    -59 

11.      -5m 

13. 

.Sxy 

+7 

+21 

+9m 

-4:.6xy 

-3 

+22 

-4m 

-.Ixy 

-11 

-2 

+7m 

5.2xy 

-17 

10.    +33a2 

12.    -45y 

14. 

a 

-25 

-18a2 

-ISy 

2a 

+40 

-21a2 

-20y 

-2a 

+43 

+6a2 

+50y 

-a 

Arts.  29,  30,  3l]     MULTIPLICATION  IN  ARITHMETIC        35 
15. 


16. 


8.5 

18. 

3(2)2 

21. 

hx 

24. 

-Ixy 

-.75 

5(2)2 

-2x 

ixy 

.05 

(2)2 

iT 

-^\xy 

-3.00 

19. 

4a 

22. 

•^xY 

25. 

a 

.42 

-8a 

-§xY 

2 

-.01 

2a 

xhf 

a 
"3 

a 
_6 

7-3 

20. 

5a6 

23. 

hn 

26. 

5.7 

-4-3 

-4a& 

\m 

-2.7 

2-3 

-ab 

-\m 

-9.7 

17. 


30.  Multiplication  in  arithmetic.  So  long  as  the  multiplier 
is  a  positive  integer,  multiplication  may  be  defined  as  the  pro- 
cess of  finding  the  sum  of  a  number  of  equal  numbers  by  the 
use  of  the  multiplication  table. 

The  result  in  a  multiplication  is  called  the  product. 

This  definition  of  multiplication  has  no  meaning  when  the 
multiplier  is  a  fraction.  Thus,  to  say  that  the  product,  f  X  |, 
means  finding  the  sum  of  |  equal  numbers  has  no  meaning. 

Hence,  an  extended  definition  of  multiplication  must  be 
>made  when  the  multiplier  is  a  fraction,  and  we  say  that  the 
product  of  two  fractions  is  the  product  of  their  numerators 
divided  by  the  product  of  their  denominators. 

31.  Multiplication  of  signed  numbers.  It  is  necessary 
again  to  extend  the  definition  of  multiplication  in  order  that 
the  products  of  positive  and  negative  numbers  shall  have  a 
meaning. 

For  example,  we  are  to  define  what  is  meant  by  such  pro- 
ducts as  3  •  -4,   -3  •  4,  and  -3  •  -4. 

We  have  said  that  3-4  =  4+4+4  =  12.  Using  the  same 
meaning  of  iimes,  we  say  that  3  •  -4  =  (-4)  +(-4)  +  (-4)  =  -12, 


36  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 

which    extends   multiphcation   to   the   case  of  multiplymg   a 
negative  number  by  a  positive  number. 

Since  to  multiply  by  +3  we  add,  it  seems  reasonable  to  say 
that  to  multiply  by  —3  we  subtract  ;  and  we  say  that  -3*4 
means  that  4  is  to  be  subtracted  3  times.     That  is, 

_3.4  =  _4_4_4  =  _12. 

Similarly,  -3  •  -4  means  that  -4  is  to  be  subtracted  3  times. 
That  is, 

-3  .-4=  -(-4)  -(-4)  -(-4) 
=  4+4  +  4  =  12. 

In  general  symbols  this  discussion  may  be  summed  up  as 
follows  : 

a  ■  h      =  ah, 
a  •  -h  =  —ah, 
-a  •  h      =  -ah, 
-a  •  -h  =  ah. 

We  may  then  state  in  words  the 

Rule  of  signs  for  multiplication.  The  product  of  two  num- 
hers  with  like  signs  is  positive,  and  the  product  of  two  numbers 
with  unlike  signs  is  negative. 

It  follows  from  the  law  of  signs  that  a  product  is  positive  if 
it  contains  an  even  number  of  negative  factors,  and  is  negative 
if   it    contains   an   odd   number   of   negative   factors.     Thus, 

-2-  3  •  -1  •  -5-  -2  =  60, 
and  -1  •  4  •  -2  •  -3  •  6  •  -2  •  -1  =  -288. 


EXERCISES   AND   PROBLEMS 
Perform  the  following  indicated  operations  : 

1.  4  •  -5.  5.   4  •  -X. 

2.  -4-5.  6.    -2-  -a. 

3.  -1  •  -2.  7.   8    0. 

4.  -7  ■  -6.  8.    -2-  -3 


Art.  31] 


EXERCISES  AND  PROBLEMS 


37 


9. 

4  ■  -5  •  G. 

17. 

8  .  -9  •  -5  •  2. 

10. 

3-0-2. 

18. 

3  -  4  •  a;  •  5  •  -0  -  -.T. 

11. 

-1  -T-  -3-2. 

19. 

(-2)2. 

12. 

-2  •  -a  ■  X. 

20. 

(-1)^ 

13. 

-2-3+5-4 

21. 

(-3)^ 

14. 

3-10  +  (-3-  - 

8). 

22. 

(-2)2.-3. 

15. 

-a ■ b • X ■  m  ■  0 

-  .T. 

23. 

(-l)-(-2)2.(-3)'\ 

16. 

-2  •  -3  -  4  -  X. 

24. 

{-ay. 

25. 

-15  =  -5  -  3  = 

5-  - 

-3. 

Find 

two  pairs  of  factors  of 

1  ; 

14;   95;    -33; 

-ab 

;  xy 

-21  ,         ,         , 

26.   If  y  =  2x  -  4,  find  the  values  of  ij  when  x  takes  on  the 
integral  values  from  -2  to  6.     Represent  the  values  of  y  by  a 
diagram. 
Solution  : 


X 

-2 

-1  1 

0  1     1  1  2  1  3  1  4  1  5  1  6 

V 

-8 

-6  1 

-4  1  -2  !  0  1  2  1  4  1  6  1  8 

The  corresponding  values  of  x  and  y  are  given  in  the  table.  The 
diagram  (Fig.  8)  is  made  as  in  Art.  18.  Notice  that  when  the  values  of 
y  are  negative  the  lines  representing  those  values  extend  downward. 


g 

_ 

7 



6 

_ 

5 

_ 

i 



3 



2 

- 

1 

_1 

_ 

-12 

-1 

0 

|l 

2 

3 

4 

5           U 

-3 

- 

I 

—  4 

—  1) 

- 

_  ~ 



-8 

- 

Fig.  8 

27.  If  s  =  t-,  find  the  values  of  s  when  t  takes  on  the  integral 
values  from  -4  to  4.  Make  a  table  showing  the  corresponding 
values  of  s  and  t,  and  repre,sent  these  values  in  a  diagram  as  in 
Exercise  26. 

28.  The  formula  C  =  ^{F  -  32),  gives  tlie  temperature, 
C,  on  a  centigrade  thermometer  in  terms  of  the  temperature, 


38  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 

F,  on  a  Fahrenheit  thermometer.  Find  C  when  F  has  the  val- 
ues -20°,  -10°,  -5°,  0°,  5°,  10°,  28°.  Make  a  diagram  repre- 
senting these  values  of  C  and  F. 

29.  liy  =  x^  -  Qx,  make  a  table  of  the  corresponding  values 
of  X  and  y,  when  x  takes  on  the  integral  values  from  —2  to  8. 
For  which  of  these  values  of  x  does  y  have  the  least  value?  The 
greatest?  For  what  values  of  a;  is  y  equal  to  zero?  Between 
what  values  of  x  does  y  increase?     Decrease? 

32.  Division  of  signed  numbers.  Division  is  the  process 
of  finding  one  of  two  factors  when  their  product  and  the  other 
factor  are  given. 

Thus,  20  H-  4  =  5,  since  5  •  4  =  20. 

The  given  product  is  called  the  dividend,  the  given  factor 
the  divisor,  and  the  factor  to  be  found  the  quotient. 


The  appUcation  of  this  definition  gives  results  as  follows: 
+7,  because   +"  •  +2  =   +14; 


7,  because  +7  •  -2  = 

-14; 

7,  because   -7  •  -2  = 

+14; 

7,  because   -7  •  +2  = 

-14. 

+14 

+2 

-14 

-2 
+14 
-2 
-14 

+2 

We  have  then  the 

Rule  of  signs  for  division.  The  quotient  of  two  nuynhers 
with  like  signs  is  positive,  and  the  quotient  of  two  numhers  with 
unlike  signs  is  negative. 

In  algebraic  symbols,  this  rule  is, 

-j-a  _     a 
+b      +6' 

—a         a 
-6  =  +6' 


Art  33]  FKACTIUxNS  39 


FKACTIUxNS 

+a 
-b 

a 
~b' 

-a 

+6  = 

a 
~b' 

33.  Fractions.  \  fraction  is  an  indicated  division.  Thus, 
f  means  3  -^  5.  Tlie  terms  of  this  fraction  are  3  and  5,  3  being 
the  numerator,  and  5  the  denominator. 

In  arithmetic  we  had  the  principle: 

The  numerator  and  the  denominator  of  a  fraction  may  be 
multiplied  or  divided  by  the  same  number  without  changing  the 
value  of  the  fraction. 

This  principle  holds  for  the  numbers  of  algebra,  and  is  of 
great  service  in  simplifying  fractions. 

EXERCISES 

Perform  the  indicated  divisions  and  check  the  results  by 
multiplication. 

1.-4.2.  7.0.5.  ^^^      ^^4.-9 

2.  -4  -  -2.         8.   0  -^  -5.  8 

3.  -15  -^  3.  9.   ab  ^  a.  _21 

14.      4-      18. 

4.  -18  ^  -2.      10.    -ab  -  a.  -^ 

5.  -7  -.  7.  11.   :^.  15.   ^^      19. 

-a  -X 

6.  19^-19.       12.    — •  16.    -^^^-     20. 

—x  xy 

Simplify  the  fractions  : 

21     -^2  23     ^^-  25     ^^• 

21-    "9    •  23.    j2  25.        ^^ 

22.    ^-  24.    'p.  26.    ^. 

15  8  ab^ 


2 
4-  5-  6 

-2-3 

2 -5+2 -6 

2 

2a  +  2b 

40  POSITIVE  AND  NEGATIVE  NUMBERS     [Chap.  III. 

or,     10  +  5  „^    i  „„    8a  +  4a 

29.   I-  32.    ^.  35.    ^^. 

36.  What  number  divided  by  2  equals  8? 

37.  Find  a:  if  I  =  8. 

38.  What  number  divided  by  -3  equals  4.5? 

39.  Find  x  ii  ~=  4.5. 

In  the  following  exercises,  the  letters  stand  for  unknown 
numbers.     Find  the  numbers. 


40. 

I-- 

45. 

3.  4. 

50. 

X 

41. 

f2-- 

46. 

.In  =  -.03. 

51. 

*?  =  -3. 

42. 

1  =  ^- 

47. 

il=- 

52 

1^-0.0 

43. 

I--- 

48. 

!-«■ 

44. 

53-- 

49. 

!-«■ 

MISCELLANEOUS    QUESTIONS  AND   EXERCISES 

1.  How  much  greater  is  9  than  4?  9  than  -4?  5x  than 
2x?     a  than  7? 

2.  Write  in  the  form  of  a  fraction  the  quotient  of  17  di- 
vided by  20.  Also  the  quotient  of  a  +  6  divided  by  m  ;  of  the 
sum  of  the  cubes  of  a  and  b  divided  by  the  sum  of  a  and  b  ;  of 
n  divided  by  a  number  6  less  than  5  times  n. 


Art.  3:3]  MISCELLANEOUS  (QUESTIONS  41 

3.  AVhat  is  the  meaning  of  a"?  Of  Oa?  Find  tiie  differ- 
ence between  a^  and  6a  when  a  has  the  values  -2,  0,  2,  and  3. 

4.  Write  the  following  using  exponents  :  2  •  7  •  7  •  7  •  7  ; 
3  •  77i-  n  ■  n  •  m  ■  m ;  x  ■  y  ■  y  •  z  ■  x  •  y  •  z  -z -z;  3-4-5-4-5-3-3. 

5.  How  is  the  sum  of  numbers  with  like  signs  found?  Of 
numbers  with  unlike  signs? 

6.  When  is  the  remainder  in  subtraction  greater  than  the 
minuend? 

7.  Perform  the  following  subtractions  on  a  number  scale  : 
8-9;    -4-7;    8  -  (-8)  ;    -5  -  (-6)  ;    -7  -  (-12). 

8.  What  is  the  definition  of  multiplication  when  the  mul- 
tiplier is  a  positive  integer?  To  what  other  cases  have  we  ex- 
tended the  meaning  of  multiplication? 

9.  Give  the  rule  of  signs  for  multiplication. 

10.  Find  the  following  products  by  adding  or  subtracting  : 
3-  7  ;  5-4;   -4    9;   -5-  -8. 

11.  What  is  the  absolute  value  of  -45;  34;  -34;  .07;  -f? 

12.  Define  division.     State  the  rule  of  signs  for  division. 

13.  In  Qabx,  state  the  coefficient  of  abx;  of  x;  o(  ab;  of  6a; 
of  6. 

14.  Give  two  illustrations  showing  how  algebraic  symbols 
may  be  used  to  shorten  the  statement  of  certain  rules  of 
arithmetic. 

x^  -  1 

15.  If  ?/  =  ^-5 r,  find  the  corresponding  values  of  y  when 

X  takes  on  the  values  -3,  -2,  -1,  0,  1,  2,  3. 

16.  U  xy  =  1,  find  the  corresponding  values  of  y  when  x 
takes  on  the  values  -2,  -1,  -.1,  -.01,  -.001.  Are  x  and  y 
increasing  or  decreasing? 


CHAPTER  IV 
EQUALITIES 

34.  Members  of  an  equality.  A  statement  that  one  expres- 
sion is  equal  to  another  expression  is  called  an  equality. 

The  two  expressions  are  called  the  members  of  the  equality. 
Thus,  in  5  +  3a:  =  4x,  5  +  3a;  is  said  to  be  the  left-hand 
member  and  4a;  the  right-hand  member  of  the  equality. 

35.  Identities.  There  are  many  different  ways  of  writing 
the  same  number.  Thus,  5  may  be  expressed  as  7  —  2,  8  —  3, 
or  2  +  2  +  1,  and  in  many  other  ways. 

Likewise,  an  algebraic  expression  may  be  changed  in 
form  and  still  represent  the  same  number.  Thus,  5a;  —  4  and 
2a;  +  3a:  -  4  represent  the  same  number  no  matter  what  num- 
ber is  represented  by  x. 

An  equality  in  which  the  members  merely  represent  dif- 
ferent ways  of  writing  the  same  number  is  called  an  identity. 
Thus,  the  equalities, 

5a:  -  4  =  2a;  +  3a;  -  4  (1) 

and 

a;  +  2a:  =  3a;  (2) 

are  identities. 

Since  the  members  of  an  identity  are  simply  different  forms 
of  writing  a  number,  the  members  are  equal  for  all  values  of 
the  letters  involved.  For  example,  the  members  in  (1)  and 
(2)  are  equal  no  matter  what  number  we  substitute  for  x.  Try 
a;  =  1,  2,  3,  and  10. 

EXERCISES 

1.  Show  that  2a  +  l  =  a  +a  +  1,  when  a  =  1,  a  =  5,  a  =  0, 
and  a  =  10. 

42 


Art.  36]  .   EQUATIONS  43 

2.  Show  that  Sa  +  2b  =  a  +  a  +  a  +b  -  2b  +  36,  if 

a  =  1  and  6  =  2; 
a  =  5  and  6  =  4; 
a  =  -1  and  6  =  -2. 

x^  -  4 

3.  Show  that  ^  =  x  +  2,  when  a:  =  0,  .r  =  1,  a:  =  3,  .t  =  5, 

X -2  '  >  J  , 

and  a:  =  10. 

36.  Equations.  An  equahty  in  which  the  members  are 
equal  only  for  particular  values  of  the  letters  involved  is  called 
an  equation. 

Thus,  the  equahty  a:  -  1  =  2  is  an  equation;  for,  the  mem- 
bers are  equal  only  when  x  =  3. 

EXERCISES 

1.  Show   that   7a:  -  4  =  10   when  x  =  2,   but   not   when 
X  =  1,  or  a:  =  3,  or  a;  =  4. 

2.  Show  that  2.t  +  4  =  3x  +  1  when  x  =  S,  but  not  when 
re  =  2. 

3.  Show  that  4a:  +  6  =  a:  +  3  when  a:  =  -1,  but  not  when 
X  =  1. 

Which  of  the  following  equalities  are  identities  and  which 
are  equations  : 

4.  X  =  X. 

5.  3a:  +  4a:  =  7a:  +  :c  -  1. 
Hint:    Try  x  =  2,  and  .r  =  1. 

«    a:- 

6.  -  =  X. 

X 

7.  2a:  =  4. 

8.  3a:  +  1  =  2.t  +  a-  +  1. 

9.  a;2  +  2x  =  -1. 
Hint:     Try  x  =  0. 

10.  x~  +  2x  =  X-  +  Sx  -  X. 

11.  x2  =  4. 

12.  a:2  -  3.T  +  2  =  0. 


44  EQUALITIES  [Chap.  IV. 

13.  a;  +  4  =  5. 

14.  3.1;  +  2ij  =  2>y  -y  -\-2x  +x. 

37.  Solution  of  equations.  In  an  equation,  a  letter  whose 
value  is  to  be  found  is  called  the  unknown  letter  or  simply  the 
unknown. 

To  solve  an  equation  is  to  find  values  of  the  unknown  which 
when  substituted  will  reduce  the  equation  to  an  identity. 

Such  a  value  of  the  unknown  is  said  to  be  a  solution  or  root 
of  the  equation. 

When  an  equation  is  thus  reduced  to  an  identity,  it  is  said 
to  be  satisfied.  Thus,  I  is  the  solution  of  a;  +  1  =  2  ;  for,  if 
1  is  substituted  for  a;,  re  +  1  =  2  is  satisfied. 

EXERCISES 

1.  Show  that  2  is  a  solution  of  2a;  -  3  =  1. 

2.  Show  that  3  is  a  solution  of  4.1:  +  1  =  13. 

3.  Show  that  0  is  a  solution  of  4a;  +  2a;  =  Ix. 

4 

4.  Show  that  -  =  is  a  solution  of  7a;  +  9  =  5. 

5.  Show  that  2  and  3  are  solutions  of  .x-  -  5a;  +  7  =  1. 

Solve  the  following  equations  : 

6.  a;  -  3  =  2.  8.   .t  -  4  =  2. 

7.  8  +  a;  =  4.  9.   a;  +  I  =  5. 

38.  Principles  used  in  solving  equations.  The  value  of  the 
unknown  in  an  equation  is  unchanged  by  the  following  : 

(1)  Adding  the  same  number  to  both  members. 

Thus,  if  2  be  added  to  each  member  of  a;  +  3  =  5,  we  have 
a;  +  5  =  7.     Both  equations  have  one  and  the  same  solution  2. 

(2)  Subtracting  the  same  number  from  both  members. 

Thus,  if  2  be  subtracted  from  each  member  of  a;  +  3  =  5, 
we  have  x  +  I  =3,  and  both  equations  have  2  for  the  value  of 
the  unknown. 


Art.  38]  VERIFICATION  BY  SUBSTITUTIOX  45 

(3)  MuUiphjing  both  members  by  the  same  number  other  than 
zero. 

Thus,  if  we  multiply  both  members  of  a;  +  3  =  5  by  2,  we 
have  2x  +Q  =  10,  and  2  is  the  value  of  the  unknown  in  both 
equations.  The  necessity  of  excluding  zero  as  a  multiplier  may 
be  seen  by  multiplying  by  x  each  member  of 

a:  +  3  =  5.  (1) 

This  gives  x~  +  3x  =  5x.  (2) 

When  X  takes  on  the  value  0,  equation  (2)  is  satisfied,  but  0  is 
not  a  solution  of  (1). 

(4)  Dividing  both  members  by  the  satne  number  other  than 
zero. 

Thus,  if  we  divide  both  members  of  2a:  +  6  =  10  by  2,  we 
have  X  +  3  =  5,  and  2  is  the  value  of  the  unknown  in  both 
equations. 

Division  by  zero  is  excluded  from  algebra.     Thus  ^  has  no 

meaning,  and  we  should  avoid  attempting  to  divide  both  mem- 
bers of  an  equation  by  an  expression  that  is  zero. 

To  illustrate  briefly  the  use  of  the  above  operations,  solve  the  equa- 
tion 5a:  -  9  =  6. 


jj        Solution: 

5x  -  9  =  6. 

Add  9, 

5X-9  +  9    =6  +  9 

Collect, 

5x  =  15. 

Divide  by  5, 

x  =  3. 

39.    Verification  of  solutions  by  substitution.     The  above 

operations  (1),  (2),  (3),  (4),  Art.  38,  are  useful  in  finding  values 
of  unknowns,  but  the  solution  of  an  ecjuation  is  not  complete 
until  the  value  of  the  unknown  found  has  been  substituted  in 
the  equation  to  test  the  result.  This  is  called  checking  })y 
substitution  or  verifying  the  result. 


46 

EQUALITIES 

Example. 

Solve  the  equation 

3x  +  10  =  28. 

Solution: 

3.r  +  10  =  28. 

Subtract  10, 

Zx 

+  10  -  10  =  28 

Collect  terms, 

3x  =  18. 

Divide  by  3, 

a;  =  6. 

Check: 

3  •  6  +10  =  28, 

28  =  28. 

[Chap.  IV. 

(1) 

(2) 

10.  (3) 

(4) 


EXERCISES 

Solve  the  following  equations  and  check  the  results : 


1. 

X  +3  =5. 

11. 

%  =  45  +  4ij. 

2. 

a;  -  3  =  5. 

12. 

13y  =  -by  +  36. 

3. 

x  +  3  =  -5. 

13. 

5a;  +  4  =  -2a;  +  10. 

4. 

X  -  3  =  -5. 

14. 

5w  -  4  =  3/1  +  18. 

5. 

X  +  4  =  9. 

15. 

8  -  6a;  +  12  +  8a;  =  24. 

6. 

X  -  3  =  12. 

16. 

3a;  +  5  +  2a;  -  1  =  24. 

7. 

2x  +  5  =  a:  - 

4. 

17. 

.T  +  2a;  +  10  +  a;  +  2a;  +  10  =  140 

8. 

3a;  +  7  =  .T  - 

11. 

18. 

3a;  -  3  +  4a;  -  16  =  68. 

9. 

bx  -\-%  =  2x 

-5. 

19. 

2a;  +  7  -  3a;  =  10. 

10. 

3a;  +  3  =  2a: 

-5. 

20. 

37/  +  2  =  2/  +  8. 

40.  Transposition.  By  the  use  of  the  principles  (1)  and 
(2),  Art.  38,  a  number  may  be  transposed  from  one  member  of 
an  equality  to  the  other  by  changing  its  sign. 

Thus,  in  the  example,  Art.  39,  in  deriving  equation  (3)  from  (1),  10 
may  be  subtracted  from  both  members  by  omitting  the  10  in  the  left 
member  and  entering  a   -10  in  the  right  member.     Likewise,  to  solve 

x  -  5  =  7, 
we  transpose   -5,  and  obtain 

a-  =  7  +  5  =  12. 

After  a  little  practice,  this  process  of  transposition  of  num- 
bers will  sometimes  be  used  to  advantage  instead  of  principles 
(1)  and  (2),  Art.  38.  However,  the  term  "transposing"  is  not 
very  essential  as  the  process  is  simply  that  of  subtracting  the 
number  from  each  member. 


Art.  41]    TRAX8LATI0X  OF  ENGLISH  INTO  ALGEBRA    47 

Example.     Solve  the  equation  5a;  +  4  =  14. 

Solution:  5j;  +  4  =  14.  (1) 

Transposing  4,  we  have  5x  =  10.  (2) 

Dividing  by  5,  x  =  2.  (3) 

Check:  5  ■  2  +  4  =  14, 

14  =  14. 

41.  Translation  of  English  expressions  into  algebraic  ex- 
pressions. In  order  to  give  algebraic  solutions  of  problems 
stated  in  words,  it  is  necessary  to  develop  skill  in  the  transla- 
tion of  English  expressions  into  algebraic  expressions. 

To  illustrate,  we  give  in  parallel  columns  a  few  equivalent  EngUsh 
and  algebraic  expressions,  together  with  some  statements  of  equality. 
Let  s  denote  the  length  of  a  side,  p  the  perimeter  and  a  the  area  of  a 
square. 

EngUsh  expressions  or  statements        Algebraic  expressions  or  statements 

1.  Four  times  the  length  of  a  side  1.    10  +  4s. 

added  to  ten. 

2.  The  square  of  a  side  plus  2.  2.     s^  +  2. 

3.  The  perimeter  of  a  square  equals  3.     p  =  4s. 

four  times  a  side. 

4.  The  area  of  a  square  is  equal  to  4.      a  =  s^. 

the  square  of  a  side. 

5.  The  perimeter  of  a  certain  square  5.    4s  =  s^.     (See  3.) 

is  equal  to  its  area. 

6.  The  perimeter  of  a  certain  square  6.    4s  =  4s-. 
>        is  equal  to  four  times  its  area. 

EXERCISES 

1.  A  rectangle  is  x  feet  wide  and  /  feet  long.  It  is  twice 
as  long  as  it  is  wide.  State  this  last  sentence  in  algebraic  sym- 
bols.    What  represents  the  perimeter? 

2.  If  the  sum  of  two  numbers  is  8,  and  one  of  them  is  x, 
what  is  the  other  numl)er? 

3.  If  X  stands  for  the  total  number  of  men  in  a  regiment, 
and  one-tenth  of  the  regiment  increased  by  5  are  sick,  what 
exjDression  denotes  the  number  of  sick? 


48  EQUALITIES  [Chap.  IV. 

4.  Write  the  algel^raic  expressions  for  x  increased  by  m, 
decreased  by  m,  multiplied  by  m,  and  divided  by  m. 

5.  The  numbers  1,  2,  3,  4,  .  .  .  are  consecutive  integers. 
How  much  greater  is  each  than  the  preceding  one? 

6.  If  n  represents  any  integer,  what  will  represent  the 
next  consecutive  integer? 

7.  If  n  represents  an  integer,  what  will  represent  the  next 
preceding  integer? 

8.  If  n  is  the  middle  one  of  five  consecutive  integers,  what 
will  represent  the  other  four? 

9.  The  numbers   2,   4,   6,   8,  .  .  .  are   consecutive   even 
integers.     How  much  greater  is  each  than  the  preceding? 

10.  If  n  represents  an  even  integer,  what  will  represent  the 
next  consecutive  even  integer? 

11.  If  n  is  the  middle  one  of  five  consecutive  even  integers, 
what  will  represent  the  other  four? 

12.  If  two  numbers  differ  by  10,  and  the  greater  is  x,  what 
is  the  other? 

13.  If  A  has  $100  more  than  B,  represent  the  money  of  each 
in  terms  of  x.     Do  this  in  two  ways. 

Hint:     First,  let  a;  =  JB's  money.     Second,  let  x  =  A's  money. 

14.  If  A  is  5  years  older  than  B,  and  B  h  x  years  old, 
represent  their  ages  in  terms  of  x  (o)  at  present;  (6)  in  10  years 
from  the  present  date. 

15.  Two  men  divide  $1000  so  that  one  shall  have  four 
times  as  much  as  the  other.  If  x  is  the  smaller  sliare,  represent 
the  larger  share  in  two  ways. 

PROBLEMS 

1.  A  rectangle  is  twice  as  long  as  it  is  wide.     The  perim- 
eter is  180  feet.     What  is  its  width? 

Hint:  Let  the  rectangle  be  .r  feet  wide. 

2.  The  sum  of  two  numbers  is  18,  their  difference  is  4. 
Wliat  are  the  numliers? 


Art.  41]  PROBLEMS  49 

3.  Two  men  are  to  divide  $1000  so  that  one  shall  obtain 
4  times  as  much  as  the  other.     What  should  each  receive? 

4.  Divide  the  number  90  into  two  parts  which  are  to  each 
other  as  2  is  to  3. 

5.  The  sum  of  two  numbers  is  9,  their  difference  is  15. 
What  are  the  numbers? 

6.  The  sum  of  two  numbers  is  a,  their  difference  is  b. 
What  are  the  numbers? 

7.  The  sum  of  two  numbers  is  32  and  their  difference 
is  -4.     What  are  the  numbers? 

8.  If  n  represents  an  integer,  what  will  conveniently 
represent  the  sum  of  this  integer  and  the  next  consecutive 
integer? 

9.  Find  three  consecutive  integers  whose  sum  is  48. 

10.  Find  two  consecutive  integers  whose  sum  is  193. 

11.  A  rectangle  is  10  yards  longer  than  it  is  wide.  Its  perim- 
eter is  80  yards.     Find  the  dimensions. 

12.  A  rectangle  is  three  times  as  long  as  it  is  wide,  and  the 
perimeter  is  128  feet.     Find  the  length  and  width. 

13.  The  United  States  has  56,000  more  miles  of  railway 
than  Europe.  The  two  together  have  409,000  miles.  Find 
the  mileage  of  each. 

14.  Three  boys  together  have  120  marbles.  If  the  second 
has  twice  as  many  as  the  first,  and  the  third  five  times  as  many 
as  the  first,  how  many  has  each? 

15.  A  farmer  has  three  times  as  many  hogs  as  horses,  and 
twice  as  many  sheep  as  horses  and  hogs  together.  If  there 
are  120  animals  in  all,  how  many  are  there  of  each  kind? 

16.  A  plumber  and  two  helpers  earn  together  S7.50  per  day. 
How  much  does  each  earn  if  the  plumber  earns  four  times  as 
much  as  each  helper? 

17.  The  sum  of  three  consecutive  integers  is  78.  What 
are  these  three  numbers? 

18.  A  merchant  owes  A  three  times  as  much  as  he  owes  B, 
he  owes  C  twice  as  much  as  he  owes  A,  and  he  owes  D  as  much 


50  EQUALITIES  [Chap.   IV. 

as  he  owes  A  and  B  together.     If  the  sum  of  his  indebtedness 
to  A,  B,  C,  and  D  is  $14,000,  how  much  does  he  owe  each? 

19.  The  length  of  a  field  is  3  times  its  width,  and  the  dis- 
tance around  the  field  is  200  rods.  If  the  field  is  rectangular, 
what  are  the  dimensions? 

20.  If  three  times  a  certain  number  is  added  to  twice  the 
number,  the  sum  is  35.     What  is  the  number? 

21.  A  real  estate  agent  purchased  3  houses,  paying  twice 
as  much  for  the  second  as  for  the  first,  and  four  times  as  much 
for  the  third  as  for  the  first  ;  if  the  difference  of  the  cost  of  the 
second  and  third  is  $3000,  what  is  the  cost  of  each? 

22.  A  room  is  15  feet  long,  14  feet  wide  and  the  walls  contain 
464  square  feet.     Find  the  height  of  the  room. 

23.  How  much  must  be  invested  at  6  per  cent  simple  interest 
to  amount  to  $650  in  5  years? 

24.  Two  motor  cars  can  run  one  at  20  miles  an  hour  and  the 
other  at  25  miles  an  hour.  If  the  faster  car  sets  out  to  catch 
the  slower  when  the  latter  has  15  miles  start,  in  how  many 
hours  will  it  catch  up? 

25.  A  golfer  knows  that  it  is  380  yards  from  the  tee  from 
which  he  starts  to  the  hole.  After  playing  two  strokes  with 
the  brassie  and  one  with  the  mashie,  he  finds  the  ball  5  yards 
short  of  the  hole.  Assuming  that  he  played  in  a  straight  line, 
and  that  each  brassie  stroke  is  twice  as  long  as  a  mashie  stroke, 
what  is  the  length  of  each  stroke? 


Art.  41]      REVIEW  EXERCISES  AND  PROBLEMS  51 


REVIEW  EXERCISES   AND   PROBLEMS 

1.  Give  a  factor  of  each  of  the  following  and  folate  its  coefficient : 
3mn;  ax^ ;  49xj/2;  hex-. 

2.  Write  the  following  expressions  using  exponents  :  x  ■  x  ■  x  ■  y  ■  z  ■  z  ; 
3  •  3  •  3  •  s  •  s  ;   125  ;  10000  ;  m-n-x-m-n-n-x. 

3.  Write  the  following  expressions  without  using  exponents:  2a^; 
(I)';  a¥;  3xV- 

4.  Add  on  the  number  scale  :   5  +  (  -  2)  +  3  +  (  -9)  +  2  +  (  -2)  ; 
-8  +  6  +  2  + (-5) +7  + (-6). 

5.  Subtract    on    the    number    scale  :      -4  -  (  -4)  -  6  -  1  -  (  -9) 
-(  -2)  ;  8  -  (  -8)  -  7  -  (  -2)  -  10  -  (  -1). 

6.  Show  on  the  number  scale  that    -2  -  (  +4)  =    -2  +  (  -4)  ;   and 
8  -  (  -4)  =  8  +  (  +4). 

7.  Write  a  formula  giving  the  cost  c  in  dollars,  of  40  chickens,  aver- 
aging p  pounds,  at  r  cents  a  pound. 

8.  Write  a  formula  which  gives  the  cost  c  in  dollars  of  n  miles  of  wire 
fencing  at  k  cents  a  rod. 

9.  Write  a  formula  which  gives  the  cost  c  in  dollars,  of  m  miles  of 
wire  at  d  cents  a  pound,  if  one  rod  of  wire  weighs  p  pounds. 

Perform  the  following  multiplications  and  divisions  : 

10.   -2  •  3  •  -5. 


11.  -1  •  -17  •  -2. 

12.  2-  3-  -11  ■  -1. 

13.  10    0-  -10-  2. 


14. 

-18 
2" 

18        100 

^^-   -loio- 

15. 

-4.. 5 
2 

l..'\'. 

16. 

-12- 
-2.-2 

-6 
-3' 

20.^  +  4. 

17. 

-2    0- 

3 

„,     2 -6 +  3 -12 

2-3 


22.  Define  and  illustrate  the  terms  power,  exponent,  coefficient,  abso- 
lute value. 

23.  If  a  is  greater  than  h,  is  the  absolute  value  of  a  greater  than  the 
absolute  value  of  6?     Give  examples. 

24.  Give  the  results  in  the  following  :   a  +  0;a-0;    a  ■  \  ;   a  -  \  ; 
aO;  0  ~a. 

25.  How  many  values  of  x  satisfy  2x  =  3x  -x? 

26.  IIow  many  values  of  x  sati.sfy  2x  =  2x  -  x? 

27.  Distinguish  between  an  equation  and  an  identity. 


52  EQUALITIES  [Chap.  IV. 

28.  The  equation  a;  -  5  =  1  is  satisfied  by  x  =  6.  If  4  is  added  to 
the  left  member  and  5  added  to  the  right  member,  is  the  resulting  equa- 
tion satisfied  by  x  =  6? 

29.  State  four  principles  used  in  solving  equations. 

30.  Supply  the  missing  term  which  makes  the  equality 

5x  +  6  -  X  =  14  +  ?  -  12  +  4  +  2x 
an  identity. 

31.  Find  the  corresponding  values  of  .5x  +  2  when  x  has  the  values 
0,  1,  2,  3,  4,  5.     Represent  these  values  graphically  as  in  Art.  18. 

x^  -  8 

32.  Find  the  corresponding  values  of  — --^  when  x  has  the  values 

-1,  0,  1,  2,  3. 

33.  Which  of  the  principles  referred  to  in  Exercise  29  are  used  in 
obtaining  the  second  and  third  of  the  following  equations  : 

If        f  of  A's  money  =  $400,  (1) 

then   I  of  A's  money  =  $200,  (2) 

and     f  of  A's  money  =  $600.  (3) 

34.  Upon   what   principles   does   transposition    depend?     Illustrate. 
35.   The  beam  AB  supported  at  its 

center  (see  figure),  just  balances  when 

one  end  is  weighted  with  x  +  5  pounds 

and  the   other  with   12  pounds.     If  5 

I    '    I     pounds  be  taken  from  the  left  side,  how 

^ — '  ' much  must  be  taken  from  the  right  side 

in  order  that  the  balance  be  maintained?     Hence,  x  pounds  balances  how 
many  pounds? 

36.  Suppose  the  left  pan  of  a  balance  contains  x  -  7  pounds  and  the 
right  24  pounds.  Let  7  pounds  more  be  put  into  the  left  pan.  Then  what 
must  be  done  to  maintain  a  balance?     Hence,  x  equals  how  many  pounds? 

37.  Illustrate  with  the  balances  the  processes  performed  in  solving 
(a)  X  +  9  =  12  ;   (6)  2x  -  4  =  x  +  6  ;   (c)  12  -  x  =  17  -  2x. 


TV 


CHAPTER  V 
ADDITION 

42.  Terms  of  an  expression.  An  expression  may  contain 
one  or  more  +  or  -  signs  which  separate  it  into  parts.  Such 
a  part  with  the  sign  preceding  it  is  called  a  term.  Thus,  in  the 
expression,  ab  +  2xy  -  7c,  the  terms  are  ah,  2xy  and  -7c. 

43.  Monomials  and  polynomials.  An  algebraic  expression 
of  one  term  only  is  called  a  monomial.  Thus,  2x,  bah  and  abed 
are  monomials. 

If  an  expression  contains  more  than  one  term,  it  is  called 
a  polynomial.  For  example,  ab  +  2xij  -  7c,  3x  -  3ij,  and 
a  +  h  +  c  are  polynomials.  In  particular,  a  polynomial  of  two 
terms  is  called  a  binomial,  and  one  of  three  terms,  a  trinomial. 

44.  Similar  terms.  Terms  that  have  a  common  factor  are 
called  like  or  similar  terms  with  respect  to  that  factor. 

Thus,  2a  and  7a  are  similar  with  respect  to  a  ;  —5x-y  and 
Sx~y  are  similar  with  respect  to  x'^y  ;  Qahx  and  5mnx  are  similar 
with  respect  to  x. 

^  EXERCISES 

With  respect  to  what  factors  are  the  following  pairs  of  terms 
similar? 

1.  5x,  12a;.  6.   abc,  xyz. 

2.  2mx^,  Qarnx^.  7.   a¥c,  ax^y. 

3.  -4:a¥x,  lamy.  8.    -\a-mn,  5ninv\ 

4.  5-7,  9-7.  9.   3(ja'-mH\   -a-tUnK 

5.  -3  ■4- 82,  13-82. 

53 


54  ADDITION  [Chap.  V. 

45.   Addition  of  monomials.     We  have  already  seen  that 

2a  +  3a  =  5a. 
Similarly, 

7x  +  ix  +  6x  =  17a:, 
and 

ax  +  hx  +  ex  =  (a  +  b  +  c)x. 

The  method  is  the  same  if  some  of  the  terms  are  negative. 
Thus, 

8a  +  a  -  5a  -  2a  =  2a, 
and 

am  -  hm  +  em  -  km  =  (a  -  b  +  e  -  k)m. 

These  examples  illustrate  the  principle  that  the  sum  of  like 
terms  is  the  product  of  their  common  factor  by  the  sum  of  its 
coefficients. 

EXERCISES 

Name  the  common  factor  and  find  the  sums  of  the  following  : 


1. 

12a  and  4a. 

13. 

26,  96,  and  -76. 

2. 

5y  and  y. 

14. 

Sd,  0.7d,  and  d. 

3. 

Sx  and  21a:. 

15. 

1.6m,  3m,  and  -3.5m. 

4. 

-9x  and  14a;. 

16. 

ax,   -bx,  and  -ex. 

6. 

-5r  and  5r. 

17. 

am,   -2m,  and  m. 

6. 

20r2  and  -7r\ 

18. 

a6c,  xyc,  and  -2xyc. 

7. 

0.5m  and  2.3m. 

19. 

2z\  \0^,  and  -122^ 

8. 

3  •  7  and  9  •  7. 

20. 

r2,  ir2,  and  2rl 

9. 

4  -25  and-4-25. 

21. 

27rr,  97rr,  and  -127rr. 

10. 

G-  -3  and  -10-  -3. 

22. 

5   7        ,3 
G'  6'  ^"^^  6- 

11. 

-2n^  and  -7n^. 

23. 

3    n        ,  2w 

7'  T  '^""'^  r 

12. 

8 -32  and  -3-3-'. 

24. 

4    2        ,  -5 

-,  -,  and 

n    n            n 

25. 

^(x  +  y),   -7ix  +  y), 

and 

mx  +?/). 

Art.  45]  EXERCISES  55 

26.  30(5a  -  4),   -9(5a  -  4),   -27(5a  -  4),  and  11  (5a  -  4). 

27.  a{x  +  y)  and  h{x  +  y). 

28.  a{c  -  4),  7(c  -  4)  and  -3(c  -  4). 

29.  Let  xy  be  represented  by  one  unit  on  the  .scale.  Atld 
on  the  scale  ^xy,  -lOxy,  Sxy,  and  -xy. 

30.  Find  the  sums  in  the  first  five  exercises  by  adding  on 
the  scale. 

31.  How  much  richer  am  I  at  the  end  of  the  day  than  at 
the  beginning,  if  (a)  I  earn  3a  dollars  and  spend  2a  dollars? 
(6)  I  earn  2a  dollars  and  spend  3a  dollars? 

32.  One  city  lot  contains  IQx^  square  feet,  a  second  lot  is 
3x  feet  long  and  9a:  feet  wide,  and  a  third  lot  is  100  feet  long  and 
40  feet  wide.  How  many  square  feet  in  the  three  lots?  How 
many  square  feet  if  x  =  12? 

33.  If  t  =  10,  what  are  the  values  of  f,  t\  t\  t\  t\ 
and  f'l 

34.  Find  the  value  oit^  -\- 1^  +  f  +  t  +  l,\it  =  10.  Also  the 
value  of  3«*  +  6^^  +  5^2  +  2^  +  7,  and  of  9«^  +  9^^  +  9. 

35.  We  may  write  427  =  400  +  20  +  7  =  U^  +  2f  +  7,  if 
t  =  10.  Write  in  terms  of  t  the  numbers  8699,  2941,  501003, 
and  345678. 

I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I    I 

-12-11-10-9  -8    -7  -G   -5  -4   -3  -2    -1      0      1      2      3      1      5      6      7      8     9     10    11    12 

Fig.  9 

36.  Show  on  the  number  scale  that  5  4-3  =  3-1-5.  Show 
that  a  +  h  =  h  +  a  when  a  and  b  have  the  sets  of  values  4,  7  ; 
-3,  8;  -5,    -6;   10,  -13;    -12,  12. 

37.  Show  on  the  number  scale  that 

(3  +  5)  -H  2  =  3  -H  (5  +  2). 

38.  Show  that  (a  +  6)  +  c  =  a  +  (b  +  c)  when  a,  b  and  c 
have  the  sets  of  values  1,  4,  7  ;  6,  -3,  8  ;  -3,  -7,  -2  ;  -5, 
8,  -3. 


56  ADDITION  [Chap.  V. 

46.  Simplifying  polynomials.  The  terms  of  a  polynomial 
may  be  arranged  and  grouped  in  any  manner  without  changing 
the  value  of  the  polynomial.* 

By  the  use  of  this  principle  we  are  able  to  reduce  many 
polynomials  to  simpler  form.  The  result  in  the  following  ex- 
ample is  found  by  first  rearranging  the  terms  and  then  adding 
together  like  terms. 

Sx  +  2y  -  3x  +  Qz  -  7y  +  'Sz  =  8x  -  3x  +  2y  -ly  -\-Qz  +  2>z 
=  bx  -  ^y  -{-  92. 

EXERCISES 

Simplify  : 

1.  X  +  4i/  -  2x  +  Sx  +  6?/  -  y. 

2.  3a  -  4  +  8  -  6a  -  a  +  1. 

3.  2.5a^  -  7a  +  Sa^  -  Z.ba?  -  a?  +  la. 

4.  A  -  B  +  C  -  2A  +  35  -  C. 

5.  3/^2  _  bn-  +  7m2n2  +  Sn^  -  m^. 

6.  4  •  5  -  8  •  52  +  3  •  52  -  5. 

7.  3-8  +  4-7-8-7-3-8. 

8.  \a  +  Im  -  4:k  +  Ik  -  |a  +  \m. 

9.  .24r2  +  35r  +  .6r^  -   .Ir. 

^"  ,  ^ 

10.  ni-^-n  +  ^- 

11.  Qxhj  +  3xy^  -  xhj  -  ixy-  +  7xif. 

12.  87rr^  +  57rr-  -  bwr'^  +  7rr-. 

13.  3  •  11  -  5  •  5  +  6  •  5  -  9  •  11  -  5  +  11. 


*  This  principle  includes  two  fundamental  laws  of  addition  : 

(1)  The    commutative    law,    or    law    of    order,    which    says    that 
a  +  b  =  b  +  a. 

(2)  The    associative  law,    or  law    of    grouping,   which    says    that 
a  +  b  +  c  =  a  +  (b  +  c). 

Thus,  8  +  5  -  .3  -h  11  =  5  +  (11  +  8  -  3)  =  (-  3  +  8)  +  (11  +  5)  =  21. 
Similarly,  2a  +  6a  -  15a  =  2a  +  (6a  -  15a)  =  ~15a  +  (2a  +  6a)  =  -7a. 


Arts.  47,  IS]        ARRANGEMENT  OF  TERMS  57 

14.  ha  -  lb  +  lc  -  \a  +  lb  -  \c. 

15.  5a6  +  8ac  -  Ibc  -  2ab  +  9ac  +  2bc  -  2ab. 

16.  a-'ft-"  -  3a2''6"  +  ab"  -  5a"6="  +  lab"  -  a^-^b". 

17.  7)1''  -  271^  +  5w^  +  3/n^  -  4m~  +  if-  -  7i'. 

18.  2(a  +  6)  +  5(a  +  b). 

19.  3(.r  -  ?/)  +  9(c  -  rf)  -  (x  -  ?/)  -  4(c  -  d). 

20.  (m  +  n)"  -7(/H  +  ^0  +  (wi  +  7\)  +  6(m  +  ?i)^ 

47.  Arrangement  of  terms  in  a  polynomial.  The  polynomial 
a^  +  3a:-  +  3x  +  1  is  said  to  be  arranged  according  to  the  de- 
scending powers  of  X  ;  1  -  ?,y  -{-lif  +  Qy^  is  arranged  accord- 
ing to  the  ascending  powers  of  y  ;  and  Sx^  +  4x-y  -  8xy-  +  y^ 
is  arranged  according  to  the  descending  powers  of  x  and  the 
ascending  powers  of  y. 

In  the  addition,  subtraction,  multiplication,  and  division  of 
polynomials  which  contain  different  powers  of  the  same  letter, 
it  is  usually  advisable  to  arrange  them  according  to  the  ascend- 
ing or  descending  powers  of  one  letter. 

48.  Addition  of  polynomials.  In  the  addition  of  polynomi- 
als it  is  convenient  to  write  the  terms  in  columns,  each  column 
containing  only  like  terms. 

Example.     Add  2x  +  y  +  5z,  7x  -  y  -  2z,  and   -4x  +  3y  -Iz. 
Solution:  Check:  Let  x  =^\,  y  =  \,z  =  \. 

2x  +    2/  +  52  8 

7x  -    y  -2z  4 

-4x  +  3y  -  72  z3 

5x  +  3?/  -  42  4 

A  check  on  an  operation  is  another  operation  that  tests  the 
correctness  of  the  first. 

The  above  solution  may  be  checked  by  substituting  any  set 
of  numbers  for  the  letters.  Let  x  =  \,  y  =  \,  and  z  =  \.  The 
values  of  the  given  polynomials  are  then  8,  4,  and  -8,  and  the 
sum  of  these  values  is  4.  Since  the  sum  thus  found  is  the  same 
as  the  numerical  value  of  the  answer  iov  x  =  I,  y  =  \,  z  =  \, 


58  ADDITION  [Chap.  V. 

the  answer  is  said  to  check.  Such  a  check,  though  not  strictly 
a  proof  that  no  mistake  has  been  made,  shows  that  the  result 
is  probably  correct.  Failure  to  check  shows  that  a  mistake 
has  been  made. 

EXERCISES 

Add  : 

1.  Sx^ly  +  92 
Sx  +  ^y  +  z 
^x  -hy  -  32 

2.  8a  -    &  +  2c  7.      l.Gm^  -    Zmn  +  3.05/1^ 
7a  +  46  -    c  -  .9/^2  +  Amn  -    Ahr? 

a  -    h  -    c  l.lm^  +      mn  -  2.2^n^ 


20x3 

-  4x^y  +  12xy^  + 

yZ 

-3^3 

+  5x'y  -  34x7/2  - 

W 

x" 

-    x'y  +      xy"^  + 

yZ 

3.  18^2  _  7a:  +  3 
-4a;2  +  X  -  9 
10x2  _  14a;  +  7 


4(a- 

-6)2- 

8(a- 

-6)  + 

15 

21(a- 

-6)2- 

ll(a  - 

-6)  - 

27 

-14(a 

-6)2  + 

(a- 

-6)  + 

4 

4. 

4A  -    35  +  20C        9. 
-9A  +  135  -  19C 
5A  -  105  -      C 

?(a-6)2-    ?  (a-6)  - 

5(x  -  6)  +  12(1/  +  9)  +  32 

(x  -  6)  -  30(y  +  9)  -    2 

-  2(x  -  6)  +    5(y  +  9)  -  82 

5. 

W  -  X  +  ?/  +  2 

w  +  x  +  y -  2 

-M)  -  X  -  1/  -  2 

10.  6(m  -  n)2  -  9(m  -  n)  +  11,         8(w  -  n)  -  27(m  -  n)2, 
43  +  (m  -  «)2  -  (w.  -  r?),  9  -  9(m  -  n)2. 

11.  x*  -  3x'^  +  x2  -  5x  +  8,    Ox"  -  6x2  ^  ^^    ^.s  _^  14^2  _  7^ 
1  -  X  +  x2  +  x^  -  x'*,   X  +  2. 

12.  3a6  -  46c  +  cd,      -a6  +  Gcrf  +  126c,      a6  -  6c, 
a6  +  6c  +  cd. 

13.  6x3  _  42-2^^  _^  ^^^1  _  yz^  _2a;3  ^  ^y?^      5^2^^  _  i2x?/  +  Zy^, 
14x3  _    ia.2^  ^  3.^2  _    3y3 

14.  \r  -\8^  t,  2t  -  4ts  -  r,  2^s  -  ^t,  Sr,  -^s,  t. 


Art.  48]  EXERCISES  59 

15.  27a'  -  4a2  +  a  -  10,  2a  -  12  +  9a^  -  a',  18  +  5a', 
6  -  3a3  +  2a2  -  9a,    a  -  2. 

16.  5(fc  +  Z)2  -  2{k  +  0-7,  20(^-  +  /)2  _  (k  +  1)  +5, 
-6(fc  +  0'  +  3(A;  +  0-9. 

17.  3"  -  3»-i  +  6,  5-3"  +  4-3"-i  -  1,  -2-3"  -  4-3"-i  +  8. 

18.  ^a  -b  +  Ic,  2a  +  fb  -  ^c,  a  -  b  -  c,  -fc  +  f 6  -  4a. 

19.  7t'  +  4t*  -f  +  3t--t  +  l,       V~  -  t'  +  t% 

ZP  -t  +  Z2  -  8f,        t'  +  1,  17  -  t  -  P  -  t'  -  t^  -  tK 

20.  a  +  .16  +  .01c  +  .OOlrf,         rf  -  .Ic  +  .016  -  .001a, 
.la  -  6  +  c  -  Ad. 

21.  a  +  6,  c  -  d,  d  -  a,  a  +b  -  c  +  d,  b  +  d,  -  a  -  c, 
c  —  d  +  b,  a,  b,  ab,  c. 

22.  (a  +  b)  +  (c-d)  -  (e  +/),  5(a  +  6)  -  ll(c  -  rf)  + 
8(e+/),     (c-rf)  -2(a  +  6)  -14(e+/). 

23.  Am  +  .2w  +  .3p  +  .4g,  .Olg  -  .02m  +  .03p  -  .04/1, 
.OOln  -  .002p  -  .003?  -  .004m,  .085m  -  .167p  -  .225g. 

24.  Find  the  sum  of  P  and  3  times  Q,  when 

P     =     X'+    Zxhj     +    3X1/2     ^    yZ^ 

and 

Q  =  2x'  +  bxhj  -  4x?/2  -  if. 

25.  Use  the  same  values  of  P  and  Q  as  in  the  last  exercise, 
and  find  the  value  of  2P  —  Q  when  x  =  \,  and  y  =  2. 

26.  Write  236,  327,  and  413  in  terms  of  t  =  10,  and  add. 
(See  Exercise  35,  Art.  45.) 

27.  Write  8964,  50231,  100000,  847,  and  2140  in  terms  of 
t  =  10,  and  add. 

28.  Add  :   a'"  +  3a'"b"  +  3a"62"  +  b'",        8a'"  -  Sb'", 

Sa'"  -  12a-"b"  +  6a"  ¥"  -  ¥",    -  a'"  +  9a2«6«  -  27a"62»  +  27b'\ 

29.  How  much  is  the  sum  x  +  y  changed  (a)  by  increasing  x 
by  1,  2,  3,  4,  and  so  on  to  10?  (6)  By  also  increasing  y  in  the 
same  way?     (c)  By  increasing  x  and  decreasing  y? 

30.  Answer  the  same  questions  as  in  the  last  exercise  for  the 
difference  2x  -  y. 


CHAPTER  VI 
SUBTRACTION 

49.  Subtraction  of  monomials.  As  shown  in  Art.  28,  a 
number  a  may  be  subtracted  from  a  number  b  by  changing  the 
sign  of  a  and  adding  the  result  to  h. 

Example  1.  From  lOx  take   -6a;. 

Solution:  lOx  -(  -6x)  =  lOx  +  (  +6x)  =  16x. 

Example  2.  From   -8a  take   -4o. 

Solution  :  -8a  -  (  -4a)  =   -8a  +  4a  =  -4a. 

Example  3.  Perform  the  subtraction   -ixy 

Solution:      -4x2/  -  (  +i^y)  =  -'^^V  -  i^V  =  -  V^2/- 

EXERCISES 

Subtract  : 

1.  13x  7.      14s  13.    l{x  -  y) 

4a:  -8s  -{x  -  y) 

2.  20a6  8.         ahc  14.   7  •  3^  •  5^ 
-bah                          -4a6c  2  •  3^  •  5^ 


3. 

-35x2y 

9. 

-X 

15. 

11-  10 

bx^y 

9x 

-6-  10 

4. 

-12.1-3 

10. 

-u^v 

16. 

7(.r  +  a) 

-42x« 

IQll^V 

26(x  +  a) 

6. 

-8mw 

11. 

im^ 

17. 

-mp'-    q') 

-4mn 

|m2 

-20(p'  -  en 

6. 

12a262 

12. 

7.5xyz 

18. 

2  •  3  ■  5 

-Ua^-" 

-O.Zxyz 

3-5 

Arts.  49,  50]     SUBTRACTION  OF  POLYNOMIALS  61 

26.  X  +  ly  33.       a 

27.  1  34.    5x* 

a_  3^ 

28.  7x  35.   Sab 

_y  8a_ 

29.  9y  36.       0 

30.  I2k-  37.   )u  -  n 
k  in 

31.  23  38.    -r  -  s 

2-  5 


19. 

12  •  hm 
-12 -Sw 

20. 

a  +  6 
6 

21. 

3.r  -  ?/ 

—  .r 

22. 

5a  -  m 

-  m 

23. 

a  -  X 

24. 

6x  +  5 
10 

25. 

3  -a: 
-5 

32.   5^  39.   2- 3- 5 

3^  2-3 

50.  Subtraction  of  polynomials.  In  the  subtraction  of 
polynomials  like  terms  may  well  be  written  in  the  same  vertical 
column.  If  several  powers  of  one  or  more  letters  occur,  it  is 
convenient  to  arrange  the  minuend  and  subtrahend  according 
to  the  powers  of  a  particular  letter. 

Example.     Subtract  Sx^  -  7  -  bx^  -  x  from  12x'  +  Sx^  _  4x  +  2. 
Solution:  Check.     Let  a;  =  1. 

12x3  +  8x2    -4x  +  2  18 

3x3  -  Sx"    _    a;  _  7  _io 

9x3  ^  133-2  _  3x  +  9  28 

Subtract:  EXERCISES 

1.  10^2  -  3x  +  12  Z.   2x  -    ^y  -    z 

8x2  4.  2a;  -    9  a:  -  lly  +  hz 

2.  5x  +  4?/  -  3w  4.   3x3  _  4^2,,  _  3^3 
2x  -  3y  +  9u  3x^  +  4x'-y  -  8y^ 


62  SUBTRACTION  [Chap.  VI. 

5.  a  -  26  +  3c  -  4d  7.       a  -h  -1 
-a  +  3b  -  5c  -  8d  -  a  +  b  +  2 

6.  I2a6  -    Zed  +     5ef  8.   6{x  +  y)  -  2{m?  -\- n^)  +  u 
-a6  +  lied  -  50ef  2(x  +  tj)  -  4(w^  +  n^)  -  u 

?{x  +  y)  +  ?(m2  +  n2)  +  ? 

9.   27 (a  +  by  +  5(a  +  6)  -  9 

(g  +  by  -  5(a  +  6)  +  I 

10.        (/?i  +  nY  +  3(m  +  ny  +  3(wi  +  w)  +  I 
-  {m  +  n)^  +  2(m  +  n)'  -  2(//j.  +  n)  -  1 

11.  From  6r''  -  y^  -  Zx^-y  -  ^xtf  take  6.r?/-  -  2,y^  +  2x3  _  9_j.2^_ 

12.  From  7a«  -  6"  +  5cp  +  llrf^  take  rf^  -  5cp  +  5a«  -  26'^. 

13.  From  8(a  +  6)  +  12(x  +  i/)  -  2  take  5(a  +  6)  - 
4(a:  +  y)  +  102-. 

14.  From  6(r  +  s)^  -  7(r  +  s)  +  25  take  5(r  +  sy  -  (r  +  s). 

15.  From    9(x  -  I)  +  5(y  +  2)  -  4(2  -  3)     take 
9(a:  -  1)  -  2,{y  +  2)  -  9(2  -  3). 

16.  From   10(a  -  fe)  +  16(c  -  rf)  +  e  take  6e  -  4(a  -  6). 

17.  Subtract  Ix;^  -  Zy^  -  2xhj  -  7xi/  from  7y^  -  x^  +  xy^ 
-  xHj. 

18.  Subtract  2,ah  +  2a?b''-  from  -  lab. 

19.  Subtract  2(a;  +  ?/)  +  5(x  +  ?/)^  +  14  from  10  +  ~~^- 

20.  Write  8639  and  27941  in  terms  of  ^  =  10,  and  subtract 
the  first  from  the  second. 

21.  From  5  •  7'^  -  4  •  7^  +  10  •  7  -  7  take  4  •  7^  +  7^  -  6  •  7. 
Simplify  the  result. 

22.  From  2  •  21  +  3  •  31  +  4  •  41  take  5  •  41  -  6  •  21  +  31. 
Simplify  the  result. 

23.  How  much  greater  is  x^  +  x^  -{-  x^  than  x^  -\-  x^  +  x? 

24.  How  much  greater  is  x^  than  a:^?  Answer  this  ques- 
tion when  X  has  the  values  4,  -3,  0,  and  |.  What  is  the  mean- 
ing of  a  negative  answer  here? 


Aht.  50]  EXERCISES  63 

25.  Find  a  value  of  x  that  will  make  (jx^  equal  to,  one  that 
will  make  it  less  than,  and  one  that  will  make  it  greater  than 
5x3. 

26.  How  much  greater  is  0  than  -11?     Than  3x  -  40? 

27.  From  the  sum  of  5a^-  -  22ab  -  3862  ^nd  42a-  +  19ab  take 
35a2  +  27ah  -  506-. 

28.  From    5m^  -  Mm'-  +  .7m  -  .001   take 
AQm^  -  .4m2  -  3.2m  -  .31. 

29.  What  must  be  subtracted  from  lb  -  ^b  +  f  c  so  that  the 
remainder  is  fa  +  |6  -  |c? 

30.  What  subtracted  from  22  will  leave  -x^+  4x^  -  12x2  _  3.? 

31.  What  subtracted  from  x  +  y  +  z  will  leave  lOx  +  lOOy 
+  IOOO2? 

32.  From  x"  +  Qx"-hj  -lSx"--y-  +  ly^  take  ^/^  +4x"-^i/  -  Sx" 
-  x"-V- 

33.  From  3  •  2"  -  6"  +  7  •  4"  +  9  •  5"  take  2"  -  5  •  4«  -  5". 

34.  Take  x-  -  3?/2  +  Ixy  -  4  from  the  sum  of  3x^-2/2  +  13 
and  x?/  -  x2  -  ?/2  _  4. 

35.  Take  2k'^  +  bh-  +  Uhk  -  3h  +  ik  from  the  sum  of 
/i  +  4A;  -  /i2  -  A;2  +  1  and  k''  +  bh?  -  hk  +  3. 

36.  Take  the  sum  of  m^  -  Sm^n  +  Zmn^  -  n^  and 

m^n  +  -^ 71^  +  3  from  3m^  -  3n?. 

37.  Take  the  sum  of   -  2p*  +  5q*  +  r*  -  pqr  +  1, 

7pqr  -  3p*  +  (f  -  r'^,  and  p^  +  5'*  +  pqr  -  4  from  the  sum  of 
5p^  +  3g*  -  r^  and  5^  +  pqr  +  p"*  +  3. 


CHAPTER  VII 

PARENTHESES 
51.    Removal  of  parentheses.     The  expressions 

6  +  (7  +  4)  and  6  +  7  +  4  are  equal. 

Similarly,  9  +  (6  -  2)  =  9  +  6  -  2. 

In  the  same  way,    a  +  (b  -  c)  =  a  +  b  -  c. 

These  examples  illustrate  the  principle  that  the  value  of  an 
expression  is  not  changed  by  removing  parentheses  preceded  by  a 
plus  sign. 

In  subtracting  the  sum  of  3  and  5  from  12  we  may  write 

12  -  (3  +  5)  or  12-3-5, 

since  we  get  the  same  remainder,  4,  in  each  case.  In  subtract- 
ing the  difference,  7-2,  from  19  we  have 

19  -  (7  -  2)  =  19  -  7  +  2, 

since  the  value  of  each  expression  is  14. 

In  general,  such  an  expression  as  a  —  (6  —  c),  means  that 
6  -  c  is  to  be  subtracted  from  a.  Hence,  Art.  27,  we  change 
the  signs  of  the  terms  of  6  -  c  and  add. 

This  gives, 

a  -  {b  —  c)  =  a  -\-  {-  b  -\-  c)  =  a  —  b  +  c. 

These  examples  illustrate  the  principle  that  parentheses  pre- 
ceded by  a  minus  sign  may  be  removed  by  changing  the  sign  of  each 
term  within  the  parentheses. 

Expressions  often  occur  with  more  than  one  pair  of  paren- 
theses.    When  one  pair  occurs  within  another  pair,  brackets 
and  braces  may  be  used  to  avoid  confusion  (Art.  15).     All  pa- 
64 


Art.  51]  REMOVAL  OF  PARENTHESES  65 

rentheses  may  be  removed  by  first  removing  tlie  innermost  pair 
according  to  the  principles  stated  above;  next  the  imiermost 
pair  of  all  that  remains;  and  so  on. 

Example.     Remove  the  parentheses  in 

2a  -  {66  -  2c  +  [a  -  (6  -c)]  +  2h\. 
Solution: 

2a-  {66-2C+  [a-  {h-c)\+2h\    =  2a  -  {66  -  2c  +  [a  -  6  +  c]  +  26} 

=  2o-  {66-2c  +  a-6  +  c  +  26} 

=  2a  -  66  +  2c  -  a  +  6  -  c  -  26 

=    a  -  76  +  c. 

Check:     Let  a  =  1,  6  =  2,  c  =  3.     Then, 

2  -  { 12  -  6  +  [1  -  (2  -  3)]  +  4 }  =1-14  +  3. 
-10    =     -  10. 

EXERCISES 

Remove  the  parentheses  and  simplify  the  results  by  collect- 
ing like  terms : 

1.  3a  +  (2a  -  4). 

2.  5x  +  (7  -  x). 

3.  2x  +  \  -  {x^-  7). 

4.  6  -  (2a  -  36)  +  a. 

5.  n  -  1  -  (1  -  w). 

6.  n  -  1  -  (n  +  1). 

7.  a  +  h  -  c  -  {c  -h  -  a). 

8.  2x  +  (3a;  -  bz)  +  (52  -  2x  +  3y). 

9.  1  -  jl  +  [1  -  (1  +  1)  -  1]  +lj  _  L 

10.  (3a  -  46)  -  (a  -  6  +  2). 

11.  x  -  [2  +  x  -  {Zx  -  7)]. 

12.  m  -  j3n  4-  [2m  -  n  -  (5m  -  w  +  6)] }. 

13.  G3  -  [30  -  (15  -  4)  +  2]  -  1. 

14.  4x  -  [5x  +  {x  -  y)  -  y]  +  Ay. 


66  PARENTHESES  [Chap.  VII. 

15.  12xy  +  3if  +  [Qax  -  {2}f~  +  lax)]. 

16.  (2X-7/  +  7)  -  [x-{y-2)]. 

17.  4a  +  36  -  [x  +  a  +  6  -  2y  -  {x  +  y)]. 

18.  a-\-m-  j  [6  +  4a  -  (6//i  +  2)]  -  {Im  -  4a  -  3) ) . 

19.  a-h  -  [{a  -h)  -  [a  -  h  -  {a  -  h)  -  {h  -  a)]  -  h  +  a]. 

20.  p  +  (/  -  r  -  [(p  -  g  +  r)  -  (  -p  +  5  -  r)]. 

Solve  the  following  equations  : 

21.  2x  -  {\  +x)  =1. 

22.  a  +  (2a  -  1)  =  5. 

23.  (32  -  4)  -  2  =  2  +  6. 

24.  a:  -  24  +  (3x  -  9)  -  15  =  0. 

25.  (6m  -  3)  =  9  -  (4?n  +  12). 

26.  .2x  -  14  -  (x  -  12)  =  6. 

27.  4t  -2  =  5  -  {St  +  7). 

28.  .5x  -  {mx  +  9)  =  40. 

29.  Show  that  x  —  {m  +  w)  =  x  —  m  -  n  when  a:,  m,  and  w 
have  the  values :  5,  1,  2  ;  2,  9,  4  ;  0,  3,  ^  ;    -2,  2,  5. 

30.  Using  the  same  sets  of  values  show  that 

X  -  (m  -  n)  =  X  -  m  +  n. 

31.  Using  again  the  same  sets  of  values  show  that 

a:  -  (m  +  n)  is  not  equal  to  a;  -  m  +  n. 
For  what  particular  value  of  n  is 

X  -  {)n  +  n)  =  X  -  m  +  n? 

52.  Insertion  of  parentheses.  Terms  may  be  inclosed  in 
parentheses  with  or  without  changing  their  signs  according  as 
the  sign  before  the  parenthesis  is  minus  or  plus. 

Examples.  x  -2  +  b  -  y  =  x  -  (2  -  b  +  y). 

x  +  3-b  +  a  =  x+{3-b  +  c). 

That  those  results  arc  correct  may  bo  shown  by  removing  the  pa- 
rentheses. 


Art.  52]  EXERCISES  67 

EXERCISES 

In  each  of  the  following  expressions  inclose  the  last  three 
terms  in  parentheses: 

1.  a  +  b  -  c  +  d.  7,    -  X  -  y  -  z  -  w. 

2.  ab  +  pq  -  rs  -  xy.  8.   a  +  b  -  c  -  d. 

3.  4  -  2a  +  b  -  y.  9.    1  -  4m^  +  4mn  -  n^. 

4.  a-  -  b~  +  2bc  -  c^.  lo.   a  +  b  +  c  -  x  -  y  -  z: 

5.  4.r2  -  7/2  -  4?/  -  4.  11.   36  -  9m'  -  (jmhi  -  1. 

6.  x~  -  2xy  +  y~  -  a-  -  2ab  -  b"-.  12.   x^  +  x~  +  x  +  1. 

13.  A  rectangle  is  x  +  20  units  long  and  x  +  7  units  wide. 
What  is  its  perimeter? 

14.  Three  points  A,  B,  and  C  are  in  a  straight  line,  and  B 
is  between  A  and  C.  If  the  distance  from  A  to  B  ism  +  n  +  Q 
inches,  and  from  B  to  .C  is  x  -  y  +  7  inches,  how  far  is  it 
from  A  to  C?  How  much  farther  is  it  from  A  to  B  than  from 
Bto  C? 

15.  Write  the  remainder  when  the  square  of  x  plus  the 
product  of  X  and  y  is  subtracted  from  the  cube  of  x 
minus  7. 

16.  By  what  amount  is  m-  greater  than  2m  -  1? 

17.  By  what  amount  is  a  -  3  greater  than  2a  -  b? 

18.  What  is  the  smaller  part  of  x^  if  a;^  +  a:  -  1  is  the  larger 
part? 

19.  By  what  amount  does  one  million  exceed 
dt'  +  5t*  +Si'  +  f-  +  7t  +  5  if  t  =  10? 

20.  What  number  is  4f  +  8  greater  than  -  47?  Find  the 
number  when  ^  =  10. 

21.  What  number  is  dt-  +  t  +  1  less  than  300?  Find  the 
number  when  t  =  10. 

22.  Write  in  the  form  of  a  fraction  the  quotient  of 
9^2  +  I2f  +  4  divided  by  St  +  2.  Find  this  quotient  when 
t  =  10. 


68  PARENTHESES  [Chap.  VII. 

23.  What  is  the  error  in  the  statement  x-y  +  z-w  = 
X  -  (ij  +  z  -  w)? 

24.  What  is  the  error  in 

ax  +  bij  -  ay  -  bx  =  {ax  -  bx)  -  {ay  +  bij)? 

53.    Collecting  literal  coefficients. 
Add  : 

1.        ax 

bx 


{a 

+  h)x 

cy 
-dy 

(c 

-d)y 

-ax 
-bx 

■  {a  +  b)x 


4. 

am 

bm 

5. 

3m 

am 

6. 

av 

-7v 

7. 

-(JX 

12. 

av^  —  Zmv 

-mx 

-3?'2  +       V 

8. 

3.t2  +  bx 
kx"  +  Qx 

13. 

gt  -  ar 
-3t-  br 

9. 

ap^  -  np 
-4p2+    p 

14. 

4x2  _  7^ 
bx""  -  az 

0. 

ax  -  by  +  cz 
-2x  -  dy  +   z 

15. 

3c2  +  2d 
l&  +  nd 

l1. 

be  -  xy  +  dy 
-26c  +  xy  -  dy 

16. 

SgP  -  hgt 
2gf+    gt 

CHAPTER  VIII 

MULTIPLICATION 

54.   Products  of  powers.     It  follows  from  the  definition  of 
an  exponent,  Art.  11,  that 

a~  =  a  •  a, 

a^  =  a  •  a  •  a  •  a, 

and  hence  that  a^-a*-=a-a-a-a-a-a  =  a^  =  a^+*. 

Similarly,  x-x^-x^  =  x-x-x-x-x-x-x-x  =  x^  =  2^1+2+5^ 

These  examples  illustrate  the  following  rule  for  combining 
exponents  in  multiplication : 

The  exponent  of  a  letter  in  a  product  equals  the  su7n  of  the 
exponents  of  that  letter  in  the  factors. 

This  rule  may  be  stated  in  algebraic  symbols  in  the  form, 

am  .  a"  =  W"  +  n.  (1) 

EXERCISES 

1.   Use  the  definition  of  an  exponent  and  show  the  meaning 
of  62  ;  c" ;  a262  ;  Qx^y  ;  SPm^n  ;  a". 

Complete  the  following  indicated  multiplications  : 


2.  a--  a^-  a^. 

8. 

(xr-. 

14. 

102 

10^ 

20.  a2  ■  a". 

3.  h'-h-  hK 

9. 

(m'-y. 

15. 

(ir- 

21.  6"  •  b'\ 

4.  m  ■  m  •  m''. 

10. 

3  •  3  •  3'". 

16. 

ar 

22.  x^"  ■  .t2«. 

5.  \f  ■  if  ■  y. 

11. 

u^y. 

17. 

ar 

23.  r-"  ■  r\ 

6.  22 .  2\ 

12. 

-z' .  z'  ■  z\ 

18. 

(!) 

(f)^ 

24.  (-ay. 

7.  -x'-x'. 

13. 

J.8  .  ,.2  .  J.19 

19. 

20- 

20^ 

25.   -a--{-ay 

26.   Verify  that  a  -h  =  h  ■  a,  when  a  and  b  have  the  values 
47,  54  ;   18,  1.3  ;  229,  11  ;  341,  2.5G. 
69 


70  MULTIPLICATION  [Chap.  VIII. 

27.  Verify  that  a  •  h  ■  c  =  a  ■  c  ■  b  =  c  ■  a  ■  b,  when  a,  b,  and 
c  have  the  values  :  3,  4,  5  ;  25,  35,  41  ;   .1,-7,  .2. 

28.  Verify  that  (a  ■  b)  ■  c  =  a  •  (b  ■  c),  when  a,  b,  and  c  have 
the  values  given  in  the  last  exercise. 

29.  What  must  be  the  value  of  m  so  that  (a)  m  •  5  •  6  =  15; 
(6)     (m  ■  6)  •  (3  •  5)  =  180  ;    (c)   .2  •  6  •  m  •  4  =  96? 

30.  Show  that  5aY'  ■  2a^'(/  =  (5  •  2)  •  (a^  •  a")  •  (y"  ■  y),  when 
a  =  I  and  y  =  2. 

31.  Show  that  -Ixhjz  ■  Sxy-z  •  -xV  =  2I.t^  y^z-,  when  x  =  1, 
y  =  2,  and  2-  =  3. 

32.  Show  that  (a"*)"  =  a"*",  when  a  =  5,  m  =  3,  and  n  =  2. 

33.  Show  that  (a"*)"  =  a""'  when  m  and  n  are  any 
positive  integers. 

Hint:  (a"*)"  =  «»».  a^  .  am  ...  to  n  factors. 

55.  Products  of  monomials.  The  factors  of  a. monomial 
may  be  arranged  and  grouped  in  any  manner  without  changing 
the  value  of  the  product.*  Thus  the  order  of  the  factors  may 
be  changed,  as  in  5  •  6  =  6-5.  Also,  the  factors  may  be  grouped 
in  different  ways,  as  in  (3  •  5)  •  8  =  3  •  (5  •  8) . 

We  make  use  of  this  principle  in  finding  the  product  of 
monomials. 

EXERCISES 

1.   Find  the  product  of  7x-  and  8a;'*. 

Solution:  7.r-  •  8.i-^  =  (7  •  8)  ■  (x-  ■  .(•■')        (Since  the  factors  may  be  re- 
arranged  and  grouped   in 
any  manner.) 
=  56.r^.  (Using  the  law  for  combin- 
ing exponents  in  multipli- 
cation.) 

*  This  principle  combines  two  fundamental  laws  of  multiplication : 

(1)  The  commutative  law,  or  law  of  order,  which  says  that 
a   b  =  b  ■  a. 

(2)  The  associative  law,  or  law  of  groujung,  which  says  that 
{a-b)  ■  c  =  a-  {b  ■  c). 


Art.  55]  EXERCISES  71 


2.   Find  the  product  of  -Sam^n^  and  Qa^m^n^. 

Solution:  -Sam-n^-  ■  Gahn^n^  =  (-3  •  6)  •  (a  •  a')  •  (m^  •  m' 
=  -18a%j-^n^ 

Complete  the  following  indicated  multiplications 


1. 

a'  •  a^x. 

19. 

(xhjzy. 

2. 

3a:2  ■  h3^. 

20. 

{Za'¥cY. 

3. 

5pq  •  -q. 

21. 

(5afy. 

4. 

3w  •  4w. 

22. 

(-Um^nyy 

5. 

xy  ■  xy. 

23. 

(Saxh/y. 

6. 

2a  ■  5b. 

24. 

{-bvWY- 

7. 

8u~v  •  -7uv^. 

25. 

4  •  W- 

8. 

12a'b'c'  ■  ^ahc. 

26. 

-5.|ml 

9. 

f  ^-2 .  InA 

27. 

„    4mn 
'■    7    • 

10. 

4m  •  2n. 

28. 

^'     3    ■ 

11. 

\u  ■  \v  •  Iw. 

29. 

11.  -5-^1 
11 

12. 

xh/z*  ■  xyz  ■  xhfz^. 

30. 

bah 
^'    2  ' 

13. 

-3a  •  -4&  •  -c. 

31. 

5 

14. 

IW-xhjz-  -6ifz'-  -ixK 

32. 

12-^^. 
-4 

15. 

2'  -S-x-  -2-  3^.T«. 

33. 

_    -3a 
^^•-13- 

16. 

43  .  72^^2^2  .  4  .  J^ji^ 

34. 

4.2^^. 
12 

17. 

5'^  •  82p5  -hip. 

35. 

--^:- 

18. 

-1.2.-3.4.-i.i--i- 

36. 

-ii- 

72 


MULTIPLICATION 


[Chap.   VIIL 


56.  Multiplication  of  a  product  by  any  number.  A  product 
is  multiplied  by  any  number  by  multiplying  any  factor  of  the 
product  by  that  number. 

Thus,  3(2-3-5)=6-3-5  =  2-9-5  =  2-315. 

5  •  {2xy)  =  \Qxij  =  2  •  5x  •  ?/  =  2.U  •  bij. 

EXERCISES 
Perform  the  following  multiplications  in  at  least  two  ways. 

10(8 -2^). 
12i(4  •5-7). 

-2(-5-0). 
_|(_i.6.-3). 
-a(-b  •  -c  ■  d). 

13.  Show  that  m(x  +  y  +  z)  = 
mx  +  my  +  mz,  when  m,  x,  y,  and  z 
have  the  values  :  2,  3,  4,  5  ;  3,  9, 
-4,  5  ;  6,  0,  -6,  -11  ;   0,  7,  2,  1. 

14.  Show  that  m{x  -  y  -  z)  = 
mx  —  my  —  mz,  when  m,,  x,  y,  and  z 
have  the  values  given  in  the  last 
exercise. 

.    We  know  that 

20  +  28  +  12. 


1. 

3(4-5). 

7. 

2. 

6(1  •  7). 

8. 

3. 

10(7  •8-2). 

9. 

4. 

12(i  •  i  •  8). 

10. 

5. 

4{3xy). 

11. 

6. 

3(12mV). 

12. 

Fig.  10 
57.    Product  of  a  polynomial  by  a  monomial 


4(5  +  7  +  3) 

This  illustrates  the  rule  that  to  multiply  a  polynomial  by  a 
monomial,  we  multiply  each  term  of  the  polynomial  by  the  mono- 
mial and  add  the  resulting  products. 


Fig.  11 


Art.  57] 


POLYNOMIAL  BY  MONOMIAL 


73 


This  principle  is  stated  in  algebraic  symbols  in  the  form 

a{b  +  c  +  (i)  =  ob  +  ac  +  ad.* 

The  product  of   two   numbers  may   be  represented  by  a 
diagram. 

Example  1.    The  product,  3    4  = 
12,  is  illustrated  by  Fig.  10. 

Example  2.     The  product, 
4(5  +  7  +  3)  =  20  +  28  +  12,  is  illus- 
trated by  Fig.  11. 

Example  3.     The  product, 
a(b  +  c  +  d)  =  ab  +  ac  +  ad,    is  illus- 
trated by  Fig.  12. 


ab 

ac 

ad 

b 

c 

d 

a (b+c+d) 

b+c  +  d 
Fig.  12 

EXERCISES  AND   PROBLEMS 

Find  the  following  products  and  simplify  if  possible  : 

1.  8(5  -t-  4  -  7). 

2.  5(7  -f  9  -  6). 

3.  3(200  +  70  +  9). 

4.  a{b  +  c  +  d). 

5.  6(3  +  6  -  m). 

6.  a^x^ia^  +  2ax  +  x^). 

7.  5m^n(8m^  -  4nin  +  n^). 

8.  -3ac(a2  -  9ac  -  c^). 

9.  \xHx^  +  2xy  -  6a^). 

10.  -^ab{-aH  +  4ax  -  12x3). 

11.  (-1)  •  (a  -  6)  +  4(3a  -  26  -  ^). 

12.  Si-x-y)  -  S(2x  +  y-2). 

13.  (-3)  •  (t  -  s)  +2t  +  Zs  -4. 

14.  6(a2  +  la  +  i)  -  9(2a''  -a -2). 
16.  6(7000  +  900  +  10  +  8). 

16.  3a"(a2"  -  2a"6»  +  ¥"). 

17.  a"+262"(a3"62  _  Qa^n-^i^'-n  +  Wa-'-'b). 

*  This  principle  is  known  as  the  distributive  law  of  multiplication. 


74  MULTIPLICATION  [Chap.  VIIL 

18.  22(1  +  2"  +  3"). 

19.  {x-ynix-yy-  {x  -  yY]. 

20.  {a  +  by[(a  +  by-Q{a  +  hy]. 

21.6(Ut)'  22.   8f?+^ 


,6  ^  Qj  \2    '  4 

Solution:    S\j  +  l'  -^j  =  -J  ^  ^  ~  ~8  ~  ^' 

no     ioA^«      76      c\  -_  _/7m      3n      llm\ 

23-    12(12-4  +3J-  ^^-  nT5-10+-FJ- 

/3a2      9a2      a^N  26.  16(|a  +  16  -  ic  -  d). 

2^-    l^TO+T+2/  27.  6(ix  +  fi/ -  i.  +  2). 


.e 


,,,5,v+^^-l! 


^n    onA^^  +  4  ,  2xy-Q      6xy  +  11 

30.  2U(^    20      +       20       +        20 

31.  Find  the  area  of  a  rectangle  whose  dimensions  in  inches 
are  36  and  66  —  7.  What  is  the  difference  between  the  num- 
ber of  square  inches  in  the  area  and  the  number  of  hnear  inches 
in  the  perimeter?     Check  when  6  =  2. 

32.  Find  the  volume  of  a  rectangular  solid  whose  dimensions 
in  inches  are  h,  2h,  and  3h  +  4.  Find  the  sum  of  the  number 
of  square  inches  in  the  surface,  and  the  number  of  linear  inches 
in  the  edges  of  this  soUd.     Check  when  h  =  10. 

33.  Illustrate  by  a  figure  that  4(5  -  2)  =  20  -  8. 

34.  Illustrate  by  a  figure  that  a{b  -  c)  =  ah  -  ac. 

35.  Show  that  (a  +  b)  {x  +  y)  =  ax  +  ay  +  bx  +  by,  when 
a,  h,  X,  and  y  have  the  values  :  1,  2,  3,  4  ;  4,  7,  2,  5  ;  -2,  3,  4,  -6. 

36.  Show  that  (m  -  n)  (p  -  q)  =  mp  -  mq  -  np  +  nq,  when 
m,  n,  p,  and  q  have  the  values :  4,  2,  7,  6  ;  1,  1,  6,  8  ;  0,  -3, 
2,-2. 


Art.  58] 


TWO  POLYNOMIALS 


75 


58.   Product   of   two   polynomials.     The   product, 
(3  +  4)  (x  +  y),  may  be  found  in  two  ways. 
(3  +  4)  (x  +  y)  =  l{x  +  ij) 
=  7x  +  ly. 
Also, 

(3  +  4)  (.T  +  y)  =  S{x  +  y)  +  4(x  +  y) 
=  3x  +  Sy  +4x  +4y 
=  7x  +  ly. 

By  using  the  second  method,  we  find  that 

(a  +  6)  (x  +  y)  =  a{x  +  ij)  +  h{x  +  ?/) 

^  ax  +  ay  +  hx  +  by.     (See  Fig.  13). 

These  examples  illustrate 
the  following 

Rule,  The  product  of  two 
polynomials  is  found  by 
multiplying  one  polynomial 
by  each  term  of  the  other  and 
then  adding  these  products. 

Example  1.   Multiply  a:  +  3  by  a:  +  7 


bx 

by 

ax 

ay 

Fig.  13 


3 

SX 

21 

<C 

X' 

IX 

olution: 

Check:    Let  x  =  1. 

X  +3 

4 

X  +7 

8 

x'^'ix 

7a; +  21 

x2  +  lOx  +  21 

32 

See  Fig.   14 

for  a    diagram  to 

p'iG.  14  illastrate  this  exercise. 

Example  2.   Multiply  2x~  +  3.i'  -  5  by  4x  -  7. 
Solution:  Check:     Let  x  =  2. 

2x2  +  3x  -    5  9 

4x-    7  1 


8x5 


12x2 
14x2 


20x 
21x 


35. 


8x3  -    2x2  -  41x  +  35 


76 


MULTIPLICATION 


[Chap.  VIII. 


EXERCISES 


Perform  the  following  multiplications  and  check  the  results 
by  substituting  numbers  for  the  letters: 
X  +  ij)  {x  +  y). 
a;  -  4)  (2x  +  3). 
a  -\-h)  {a  -  h). 
80  +  7)  (60  +  4). 
2x  +  y)  (3x  -  2y). 
6m2  +  5mn)  (m  -  n). 
x2  +  X  +  I)  (x  -  I). 
a;2  -  xy  +  2/^)  (^^  +  xy  -\-  y-). 
a  +  6)-. 
a  +  6)^ 
30  +  1)^ 
a  +  6  +  c)2. 

a6  +  a?W  +  6«)  (a^  -  V). 
Ax^  -x"^  +  x  -  I)  (2x2  -  4x  -  7). 
800  +  40  +  2)  (70  +  6). 

X*  -  2x?y  +  2x2?/  -  2x2/3  +  y^)  (x^  +  xy  +  ?/2). 
X  -  2)  (x  -  3)  (x  -  4). 
X  +  a)  (x  +  6)  (x  +  c). 
a^*  -  3a26  +  3a62  -  h^)  {a?  -  2a6  +  fe^). 
/«  -  i)  (2m  -  I). 
\x  -  \y)  {\x  +  \y). 
|x  -  f  y)2. 

5a  -  66  +  3c)  (2a  +  6  -  4c). 
a       6      c\    /a      h      c' 
2  ~  2  "^  27  V2  "^  2  ~  2, 
.2wi2  _  ^\tnn  +  w2)  (2m  -  .5/0- 
1.44a2  -  .72a6  +  .0%2)  (1. 2a  -  .36). 
3.5m2  -  Amn  +  .15^^)  {Am  -  (Sn). 
X"  -  5x"-i?/  +  x"-2/)  {x  -  y). 


X"  +  ij"y. 


Art.  58]  EXERCISES  77 

31.  What  is  the  area  of  a  rectangle  that  is  3x  +  5  units  long 
and  f.T  -  4  units  wide?     Check  when  x  =  6. 

32.  Show  that  (8/  +  3)  (7f  +  2/  +  6)  =  83  •  726  if  t  =  10. 

33.  What  is  the  difference  between  the  squares  of  two  suc- 
cessive integers,  the  smaller  of  which  is  x?  Is  this  difference  an 
odd  or  an  even  number? 

34.  Illustrate  the  meaning  of  (a  +  b  +  c)  (x  +  y)  by  con- 
structing a  rectangle  a  +  6  +  c  units  long,  and  x  -\-  y  units 
wide. 

35.  Show  by  a  figure  how  much  the  area  of  a  square  of  side 
a  is  increased  by  increasing  the  length  of  the  side  one  unit. 

Solve  the  following  equations  : 

36.  (3x  -  2)  -  1x  -  (12  -  3.t)  =  13x. 

37.  2(4  -y)  =  5y  -  13. 

38.  3(a  +  2)  -  (a  -  9)  =  1. 

39.  n  +  2(n  +  1)  +  3(n  +  2)  =  91. 

40.  7(4a:  -  3)  +  (7  -  8a:)  =  1. 

41.  How  much  is  the  area  of  a  rectangle  of  base  x  and  alti- 
tude y  changed  (a)  by  multiplying  its  base  by  2?  (6)  By  divid- 
ing both  the  base  and  the  altitude  by  2?  (c)  By  multiplying 
the  base  and  dividing  the  altitude  by  2? 

42.  How  much  is  the  area  of  a  rectangle  whose  sides  are 
X  and  y  changed  (a)  by  increasing  both  the  base  and  the  alti- 
tude by  2?  (b)  By  decreasing  both  the  base  and  the  altitude 
by  2? 

Draw  figures  showing  the  following  products  : 

43.  (x  +  y)  (x  +  y).  47.    (a  +  2y. 

44.  (x  -  4)  (2x  +  3).  48.    (30  +  l)^. 

45.  (m  +  7)  (m  +  5).  49.    (a  +  6  +  cY. 

46.  (2.T  +  ij)  {3x  -  2y). 


2x  = 

6. 

X   = 

3. 

1) 

-5(2 

•3 

-3)  = 

18, 

33- 

-15  = 
18  = 

18, 
18. 

CHAPTER  IX 
EQUATIONS  AND   PROBLEMS 

59.    Equations  involving  parentheses. 

Example  1.     Solve  the  equation  3(4.c  -  1)  -  5i2x  -  3)  =  18. 

Solution:  3(4.c  -  1)  -  5(2x  -  3)  =  18. 

Multiplying,  (12x  -  3)  -  (lOx  -  15)    =  18. 

Removing  parentheses,  12.c  -  3    -    lOx  +  15     =  18. 

Transposing  and  collecting  like  terms, 
Dividing  by  2, 

Check:  3(4   3 


In  certain  equations  higher  powers  of  the  unknown  than  the 
first  occur,  but  vanish  as  in  the  following  : 

Example  2.   Solve  the  equation  {x  -  5)(x  +  3)  -  {3x  -  4)  =  (x  -  1)^. 
Solution:  (x  -  5)(x  +  3)  -  (3x  -  4)  =  (x  -  l)^. 

Multiplying,  x^  -  2x  -  15  -  3x  +  4  =  x^  -  2x  +  1. 

Transposing  and  collecting  like  terms,  -3x  =  12. 

Dividing  by   -3,  x  =    -4. 

Check:  (  -4  -  5)(  -4  +  3)  -  [3  •  (  -4)  -4]  =  (  -4  -1)^, 

9  +  16  =  25, 
25  =  25. 
EXERCISES  AND   PROBLEMS 
Solve  the  following  equations  : 

1.  5(x-5)  =x-  9. 

2.  4(10  -  2x)  =  S(x  -  5). 

3.  3(9  -  2x)  -  5(2x  -  9)  =  0. 

4.  7(47/ -  3)  +  3(7  -  8?/)  =  1. 

5.  8(3.T  -  2)  -  7.r  -  5(12  -  3.r)  =  13a:. 

6.  6a  -  7(11  -  fl)  +  11  =  4a  -  3(20  -  a). 


Art.  59]  EXERCISES    AND   PROBLEMS  79 

7.  5(m  -  3)  +  4(17  -  m)  =  11  -  7(3m  -  6). 

8.  k  -  2(4  -  7A-)  =  'ik  -  9(2  -  3k). 

9.  80  -  QC4x  +  3)  =7x-  S{Qx  +  1). 

10.  Z(y  -  10)  +  n(2y  +  1)  =  17(y  -  12). 

11.  8x{Zx  +  2)  -  27  =  4.r(6a:  -  1)  -  147. 

12.  (5  -  3a)  (3  +  4a)  =  (1  -  4a)  (7  +  3a)  -  1. 

13.  21  -  3p(10p  +  3)  =  45  -  5p(6p  -  1). 

■       14.    (2/i  -  1)  (n  +  5)  -  1  =  (?i  -  6)2  +  (n  +  7)^. 

15.  n"-  +  (n  +  1)2  =  (n  +  2)^  +  (n  +  3)^. 

16.  4(2x  -  ^)  -  6(1  -  I 

17.  8(2x  -  f)  =  5  +  5(x  -  i)  -  4^^ 

18.  4(a  -  1)2  +  6(a  -  §)  =  (2a  -  5)^  +  302. 

19.  (x  -  1)3  +  5(x  +  2)  =  (x  +  1)3  -  (3a:  -  4)  (2x  +  7)  +  16. 

20.  27 (x  -  i)  -  114^^  +  l)  =  37a:  -  6(2.t  -  i). 

21.  A  line  is  divided  into  two  parts,  one  of  which  is  20 
inches  longer  than  the  other.  Twelve  times  the  shorter  piece 
equals  8  times  the  longer.     How  long  is  the  line? 

22.  A  man  paid  $12.50  for  2  wooden  golf  clubs  and  6  iron 
ones.  Each  wooden  club  cost  25  cents  more  than  each  iron 
club.     Find  the  cost  of  each.    - 

23.  The  value  of  31  coins  consisting  of  dimes  and  nickels  is 
S2.25.     How  many  are  there  of  each? 

24.  A  tourist  climbs  from  a  certain  point  up  the  slope  to 
the  top  of  Pike's  Peak  at  the  rate  of  2  miles  per  hour,  and 
descends  by  the  same  path  at  the  rate  of  4  miles  per  hour.  If 
the  round  trip  takes  6  hours,  how  long  is  the  path? 

25.  A  man  made  two  investments  amounting  to  $4330. 
On  the  first  he  lost  5  %,  and  on  the  second  he  gained  12  %. 
What  was  each  investment  if  the  net  gain  was  $251? 


80 


EQUATIONS  AND  PROBLEMS          [Chap.  IX. 


In  a  right-angled  triangle  the  square  on  the  hypotenuse  is  equal  to 
the  sum  of  the  squares  on  the  other  two  sides.  The  hypotenuse  is  the 
side  opposite  the  right  angle.     In  Fig.  15,  c^  =  a^  +  b-.     If  o  =  3,  and  6  =  4, 

then  c2  =  32  +  42  =  25,  and  c  =  5. 
By  transposing,  we  get  a^  =  c^  -b'^ 
and  62  =  c2  -  a^. 

26.  One  side  of  a  right 
triangle  is  5  inches,  and  the 
hypotenuse  is  2  inches  longer 
than  the  other  side.  How 
long  is  the  hypotenuse? 

Suggestion  :   Let  x  =  the  num- 
ber of  inches  in  the  unknown  side. 
Then   x  +  2  =  the  number  of 
inches  in  the  hypotenuse, 
and     (x  +  2)2  -  x^  =  25. 

27.  A  rope  that  is  8  feet 
longer  than  a  flag-pole  reaches 

from  the  top  of  the  pole  to  a  point  on  the  ground  32  feet  from 
the  foot  of  the  pole.     Find  the  length  of  the  flag-pole. 

Historical  note  on  the  equation.  The  very  earliest  mathematical 
writer  of  whom  we  know,  an  Egyptian  priest  named  Ahmes,  solved  equa- 
tions in  one  unknown.  He  lived  at  least  as  early  as  1700  B.C.  The  un- 
known he  called  "  hau"  or  heap.  One  of  his  problems  reads  :  "  Heap, 
its  half,  its  whole,  it  makes  16."     When  x  denotes  the  unknown,  the 

equation  to  be  solved  is  x  +  ^  =  16.     In  Egyptian   hieroglyphics  this 


c  o 

b 


Fig.  15 


Ji. 


^1 


n 


Ml 


Heap,  its  half,  its  whole,  it  makes  16 

equation  is  written  as  shown.  The  Hindus  used  the  word  color  to  denote 
the  unknown,  the  Europeans  early  used  the  word  res  (thing),  and  the 
Arabs  used  the  word  root  in  this  sense.  It  seems  that  next  the  initial 
syllable  of  each  word  was  used  to  denote  an  unknown.  Thus  ka  (from 
kdlakn  =  black)  meant  the  unknown.  It  was  not  until  the  time  of  Vieta 
(1540-1603)  that  a  single  letter  was  used  for  the  unknown.  The  con- 
ventional use  of  X  for  the  unknown  is  due  to  Descartes  (see  p.  181). 


Arts.  59,  60]       EXERCISES  AND  PROBLEMS  81 

28.  One  side  of  a  rectangular  field  is  40  rods.  The  diagonal 
of  the  field  is  20  rods  less  than  half  the  perimeter.  Find  the 
area  of  the  field. 

29.  The  difference  between  the  areas  of  two  squares  is  192 
square  inches.  The  side  of  one  is  6  inches  longer  than  the  side 
of  the  other.     Find  the  area  of  the  larger. 

30.  One  side  of  a  rectangle  is  15  inches  longer  than  the 
other.  The  area  of  the  rectangle  is  450  square  inches 
greater  than  the  area  of  a  square  whose  side  is  equal  to 
the  shorter  side  of  the  rectangle.  Find  the  area  of  the 
rectangle. 

31.  The  sum  of  two  numbers  is  2.  The  difference  of  their 
squares  is  80.     Find  the  numbers. 

32.  The  difference  of  the  squares  of  two  consecutive  integers 
is  41.     Find  them. 

33.  The  difference  of  the  squares  of  two  consecutive  even 
integers  is  436.     Find  them. 

34.  The  radius  of  a  circular  flower  bed  is  increased  2  feet, 

thus  increasing  its  area  88  square  feet.     Find  the  radius  of  the 

original  bed. 

22 
The  area  of  a  circle  is  irr^,  where  r  is  the  radius.     Use  tt  =  —  • 

35.  The  difference  of  the  areas  of  two  circles  is  423.5  square 
feet.  The  difference  of  their  radii  is  3.5  feet.  What  is  the  ra- 
dius of  the  larger? 

36.  Show  that  {2nY  +  (n^  -  1)^  =  (n^  +  1)2,  and  hence  that 
2w,  11^  —  1,  and  v}  -\-  \  may  be  used  as  the  sides  of  a  right  tri- 
angle. Give  n  the  values  1,  2,  3,  .  .  .  10,  and  find  the  corre- 
sponding values  of  2n,  n^  -  1,  n^  +  1.  The  formula  is  known  as 
Plato's  formula. 

60.  Equations  involving  fractions.  We  are  able  to  solve 
certain  equations  involving  fractions  Ijy  ai)plying  the  jirinciples 
of  fractions  learned  in  arithmetic;. 


82  EQUATIONS  AND  PROBLEMS  [Chap.  IX. 

Example  1.     Three-fourths  of  a  number  plus  5  equals  47.     What  is 
the  number? 

Solution  :     Let  x  =  the  number. 

Then,  ^  +  5  =  47. 

'  4 

3x 
Transposing,  ^  =  42. 

Multiplying  both  members  of  the  equation  by  4,  we  have, 

3a;  =  168. 
Dividing  by  3,  x  =  56. 

Check:  3|6  ^  ^  ^  ^^ 

47  =  47. 

Example  2.     One  third  of  a  number  plus  one  fourth  of  the  number 
equals  21.     What  is  the  number? 

Solution  :  Let  x  =  the  number. 

Then,  1+1  =  21- 

Multiplying  each  member  by  12,  4a;  +  3a;  =  252,  or  7x  =  252. 
Dividing  by  7,  x  =  36. 

Check:  12  +  9  =  21, 

21  =  21. 


EXERCISES  AND   PROBLEMS 


Solve 

1. 

1-*- 

2. 

1  =  3. 

3. 

hl-^- 

4. 

'-^- 

5. 

1  +  8  =  13 

,.  1       11 

«•   3=^  +  6=2^- 


7. 

X       1 
9      3  "  " 

8. 

^  =  . 

9. 

^-• 

10. 

1?  =  4. 

X 

Aht.  60]  EXERCISES  AXD  PROBLEMS  83 

11.5+0=9.  16.2  =  4. 

X  z 

12.  71  -  In  =  4.  17.   2'i  =  2  +  3  +  4- 

28 

13.  .5m  -  5  =  .5  -  5m.  18.   3x  =  y 

14.  ^^  +  3000  =  3028.  19.  ?  -  f  =  0.004. 
100  o       I 

16.1=1.  20.   1  +  1  =  2. 

21.  An  article  was  sold  for  $5.40,  thus  losing  one-sixth  of 
the  cost.     Find  the  cost. 

22.  The  increase  in  the  value  of  certain  farm  land  is  1^ 
times  the  value  15  years  ago.  What  was  the  land  worth  15 
years  ago,  if  it  is  now  worth  $220  an  acre? 

23.  After  a  decline  of  5  %  in  the  price  of  an  article,  it  was 
worth  $7.98.     What  was  the  value  before  the  decline  in  price? 

24.  The  difference  between  tV  of  a  number  and  .05  of  the 
number  is  20.     What  is  the  number? 

25.  A  bushel  of  corn  and  a  bushel  of  wheat  cost  together 
$1.50.  The  corn  costs  |  as  much  as  the  wheat.  Find  the  cost 
of  each. 

26.  A's  age  is  |  of  B's  age.  The  sum  of  their  ages  is  48 
years.     Find  the  age  of  each. 

27.  One-half  of  the  time  past  midnight  equals  the  time  till 
noon.     What  time  is  it? 

Hint:     Let  x  =  the  number  of  hours  past  midnight. 

Then,  -  =  the  number  of  hours  till  noon, 

and  X  +  2  =  12. 

28.  One-third  of  the  time  till  midnight  equals  the  time 
past  noon.     What  time  is  it? 


84  EQUATIONS  AND  PROBLEMS  [Chap.  IX. 

29.  Three-fifths  of  the  time  past  midnight  equals  the  time 
till  noon.     What  time  is  it? 

30.  Twice  a  certain  number  is  7  more  than  |  of  it.  Find 
the  number. 

31.  If  a  certain  number  is  diminished  by  99  the  result  is  the 
same  as  if  it  is  divided  by  10.     What  is  the  number? 

32.  The  sum  of  the  three  angles  of  a  triangle  is  180°.  In 
a  triangle  the  second  angle  is  3  times  the  first,  and  the  third  is 
^  the  first.     How  many  degrees  in  each  angle? 

33.  In  a  right  triangle  one  acute  angle  is  ^  as  large  as  the 
other.     How  many  degrees  in  each? 

34.  In  a  right  triangle  one  acute  angle  is  12°  greater  than 
the  other.     How  many  degrees  in  each? 

35.  In  a  triangle  the  second  angle  is  ^  the  first,  and  the 
third  is  twice  the  second.     How  many  degrees  in  each? 

36.  At  what  time  between  3  and  4  o'clock  are  the  hands  of 
a  clock  together? 

Suggestion:  Let  x  =  the  number  of  minutes  past  3  o'clock  when 
the  hands  are  together.      Since  the  hour  hand  travels  ti  as  fast  as  the 

minute  hand,  y^  is  the  number  of  minute  spaces  over  which  the  hour  hand 

has  passed  since  3  o'clock.     But  at  3  o'clock  the  hour  hand  was  15  minute 
spaces  ahead  of  the  minute  hand.     Hence, 

37.  At  what  time  between  5  and  6  o'clock  are  the  hands  of 
a  clock  together? 

38.  At  what  time  between  2  and  3  o'clock  are  the  hands  of 
a  clock  pointing  in  opposite  directions? 

39.  At  what  time  between  7  and  8  o'clock  are  the  hands  of 
a  clock  at  right  angles  to  each  other?    Two  answers. 

40.  If  it  takes  the  author  3  times  as  long  to  make  up  a 
problem  as  it  does  the  student  to  solve  it,  and  the  making  and 
solving  together  take  12  minutes,  how  much  time  would  be 
saved  each  by  omitting  the  problem? 


Art.  GO]  PROBLEMS  •       85 

41.  In  a  recent  election  46  less  than  two-fifths  of  the  votes 
were  cast  by  women.  Three-sevenths  of  the  women  voted  a 
certain  ticket,  thus  casting  540  votes.     How  many  men  voted? 

42.  A  man  travels  for  2|  hours  in  his  automobile.  At  the 
end  of  that  time  something  happens  to  the  engine,  and  he  has 
to  travel  at  half  his  former  speed  for  1|  hours  to  reach  a  garage. 
His  speedometer  shows  that  he  has  traveled  65  miles.  What 
was  his  speed  at  first? 

43.  The  total  operating  revenue  of  the  Pennsylvania  Rail- 
road Company  for  a  certain  year  was  $6,000,000  more  than 
I  of  the  total  operating  expenses.  The  total  operating  revenue 
was  $157,000,000.     Find  the  total  operating  expenses. 

44.  A  man  has  a  certain  sum  of  money  invested  at  7  %, 
and  fV  as  much  at  6  %.  The  income  from  the  two  investments 
is  $248.     Find  the  amount  of  each. 


CHAPTER  X 
DIVISION 

61.   Division  of  monomials.     Since  o?  •  a? 
from  the  definition  of  division,  Art.  32,  that 


a^,   it  follows 


Similarly,  since 
then, 


-7-  a"  =  a^. 


We  have  then  the  following  rule  of  exponents  for 
division: 

The  exponent  of  a  letter  in  the  quotient  equals  the  exponent  of 
that  letter  in  the  dividend  minus  its  exponent  in  the  divisor. 

In  all  cases  the  quotient  must  be  given  the  proper  sign 
according  to  the  rule  given  in  Art.  32. 

To  multiply  Zxy"^  by  Ax-y"^,  we  form  the  products  of  the  numerical 
coefficients  and  then  the  products  of  the  letters.  Hence,  to  divide  12x'y* 
by  3xi/2,  we  divide  12  by  3,  then  x^  by  x,  then  y'  by  y-,  and  obtain 


\2xh/ 
3xy^ 


-Ax-y^. 


If  there  are  factors  in  the  divisor  that  are  not  in  the  dividend,  the 
result  is  left  in  the  form  of  a  fraction.     Thus, 

18  _3.     2mW  _  _n2_ 
12  ~  2  '    Qm^nx  ~  Smx 


Arts.  61,  02]        DIVISION  OF  A  POLYNOMIAL  87 


EXERCISES 

Complete  the  following  indicated  divisions,  and  check  the 
results  by  multiplication  : 

1.8,2.  11.   iM!.  21.   t^ii^'. 

Sab  X 

2.  -42.6.  1^.=^.      22.   tfOipM!. 

3.  18«*3a.  13.^.  23.^^:- 

.2x  (a  +  1)2 

4.  30a^*-2.  14.   "^.  24.    (^"-^^^ 

,,     -22- 3- 5      „, 

5.  a:7/3  -  z.  15.     ^.^.^  •     25. 

6.  3a2  -  36.  16.    ~|', '.  ^'-        26. 

7.  X^?/   -4-  X2.  17.     -^|^3-  27.       ,  o^    ,,      ,     .X 

-2a;?/223  ((J  -  3)  (6  +  4) 

8.  3a'.  2a'.  18.^'.  28.    ^^x^. 

^ab&  -xy^^ 

9.  5-6.5.  19.   i^»  29.     4^?^. 

-4pg  -30.r2?/233 

10.    -IP- 13 -11.     20.    =^,-  30.   ^^'• 

62.    Division  of  a  polynomial  by  a  monomial.     Since 

a(b  +  c  +  d)  =  ab  +  ac  +  ad, 

it  follows  from  the  definition  of  division  that 

(ab  +  ac  +  ad)  -7-a  =  b+c  +  d 

This  illustrates  the  rule  : 

To  divide  a  polynomial  by  a  monomial,  divide  each  term  of 
the  'polynomial  by  the  monomial  and  add  the  quotients. 


S(x 

-y) 

2' 

32-5 

2 

33-5 

(- 

m)  (-n)  ( 

-pY 

n{-n)  (- 

V) 

(a 

-  3)  (6  + 

4)2 

DIVISION  [Chap.  X. 

EXERCISES 
Divide  and  check  the  results  by  multiplication: 

1.  Sx^  +  4:X  by  X. 

2.  lOij  -  15a;  by  5. 

3.  6a;2  +  8x  by  2x. 

4.  x^y  +  4x1/  by  xy. 

5.  14:ahx  -  4%cx  +  7bx  by  7bx. 

6.  2a  -  66  +  10c  by  2. 

7.  rc'^  -  a;2  +  x  by  x. 

8.  ?H^  -  2mH  +  mx"  by  -m. 

9.  48x^2/"^  -  18x-y^2  +  120xy~z^  by  6x?/2, 

10.  .Qm^nP  —  1.2wi2n  by  .2m^n. 

11.  .25a'6''c5  -  .5a^64c3  +  2.5a^b'^c^  by  .25a262c2. 

12.  6(a  +  xy  +  (a  +  x)  by  a  +  x. 

13.  12(wi  -  ny  -  9(m  -  n)^  +  6(?«  -  n)  by  3(m  -  n). 

4a2  _  6a  +  2      „  _    x^  +  x^-  x      „ 

14.    p; =  :  15.   =  I 

2  -X 


16. 
17. 


_6_c -rf-  1 


-1 

3pY  -  6p^g^  +  3pr/ 

3pg^ 


j3    4(c  -  rf)^  -  (c  -  d)  ^  ^  ^g    9g  +  66  +  3  ^  ^ 

-  (c  -  rf)  ■  '  3 

36(a  -  1)^  +  12(a  -  1)  ^  ^ 
12(a  -  1) 
2^     18a:^^y^  -  ^Oxhj  ^  ^ 
Gxy 


20. 


22. 
23. 


18(a  -  l)'^  -  30(a  -  1)^  _  ^ 

6(a  -  1) 
28(a  -  5)^  +  35(a  -  5)=^  -  7 (a  -  5) 
7(a  -  5) 


Arts.  62,  63]       DIVISION  BY  A  POLYNOMIAL 

2^    a"b'  +  a"+-b'  +  a"+'b'  ^  ^ 
o-b^ 

o^"b"  +  o~"b-"  -  n"b^"       ., 
2o.    


a%" 

26.   Select  monomial  divisors  for  the  following  and  divide  : 
7xy  -  1-i.r-  +  7x,     3a  +  6a-  -  9a^,     Sxy  +  ix'^y  -  Wxy"^. 

63.    Division  by  a  polynomial. 

Example  1.     Di\'ide  x^  +  2x^y  +  2xy^  +  y^  by  x^  +  xy  +  y^. 

Solution: 

x3  +  2x'y  +  2x2/2  +  y^     b"  +  xy  +  y" 
7?  +   x^y  +    xy^  |x  +  t/ 

x-y  +    xy"-  +  2/3 

x^y  +    xy^  +  2/3 

Check:  Let  x  =  1,  and  y  =  \.  Then  x'  +  2x^2/  +  2xy'^  +  2/^  =  6, 
x^  +  X2/  +  2/^  =  3,  and  x  +  2/  =  2.     Since  6^3  =  2,  the  solution  checks. 

Explanation.  (1)  It  is  convenient  to  arrange  the  dividend 
and  divisor  according  to  the  descending  powers  of  x. 

(2)  The  highest  power  of  x  in  the  dividend  divided  by  the 
highest  power  of  x  in  the  divisor  gives  the  highest  power  of  x 
in  the  quotient.  Dividing  x^  by  rr^,  we  get  x  for  the  first  term 
of  the  quotient. 

(3)  Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  it  contains  the  product  of  the  divisor  and  each  term 
of  the  quotient.  Hence,  we  multiply  x"^  +  xy  +  y^  by  x  and 
subtract  the  product,  x^  +  x^y  +  xy"^,  from  the  dividend.  The 
remainder,  x^y  +  xy^  +  y^,  contains  the  product  of  the  divisor 
and  the  remaining  terms  of  the  quotient. 

(4)  The  first  term  of  the  remainder,  x^y,  divided  by  the 
first  term  of  the  divisor,  x"^,  gives  the  second  term  of  the  quo- 
tient, y.  Multiplying  the  divisor  by  y  and  subtracting,  the 
remainder  is  zero.  The  division  is  therefore  completed,  and 
the  quotient  is  a;  +  2/- 


90  DIVISION  [Chap.  X. 

Example  2.     Divide  12.r''  -  26.c'  -  15^3  +  S-c^  -  4x  +  9  by  ix^  -  Iz^  + 
X-  1. 

Solution: 

12x5  -  26x4  -  15x3  +  8x2    _  4.^  +  9   14.^3  _  2x2  +  x  -  1 
12x5  -    6x*  +    3x3  _  3a;2  h^^p.  _r^y^  _  7 


-  20x*  -  18x3  +  11x2  -  4a. 

-  20x4  +  10x3  -    5x2  _|.  5a; 

-28x3  +  16x2  _  9x  +  9 
-28x3  +  14x2  _  7x  +  7 

2x2  —  2x  +  2  =  Remainder 

EXERCISES 
Divide  and  check  the  results  by  substituting  numbers  for 


the  letters : 

1. 

c?  +  lab  +  6-  by  a  +  6. 

2. 

x^  -  2xy  +  y^  hy  X  -  y. 

3. 

x^  +  5x  +  Qhy  X  +  S. 

4. 

m2  -  7m  +  12  by  tn  -  4. 

5. 

y^  -  y  -20hy  y  -  5. 

6. 

a2  +  2a  -  35  by  a  +  7. 

7. 

a:2  +  9  by  a;  +  3. 

8. 

a;2  +  Qax  +  80,2  by  x  +  2a. 

9. 

a2  -  9ab  +  Ub'^  by  a  -  2b. 

10. 

6a2  +  7am  +  2}n^  by  2a  +  m. 

11. 

20a;2  -  7x1/  -  Sif  by  5a:  -  3?/. 

12. 

(j2      _     ^2     jjy     (J     -     6. 

13. 

4x2  -  25y^  by  2a;  +  5y. 

14. 

a^  +  da%  +  3a62  +  b^  by  a  +  b. 

15. 

m^  +  5m2w  -  24^2  by  m^  -  3n. 

16. 

4a''  +  4a2  -  29a  -  21  by  2a  -  3. 

17. 

8a'  +  12a'b  +  Qab'  +  b' by  2a  +  b 

18. 

12w3  -  23u^v  +  Guv""  +  ryv'  by  4m  - 

19. 

0,3  _  53  by  a  _  6. 

20. 

27wi''  +  Sn^  by  Sm  +  2n. 

5v. 


Art.  G3]  EXERCISES  91 

21.  x^  —  y^  by  .r  -  y. 

22.  x^  +  y^hy  x  +  y. 

23.  p^  +  5^-  by  p^  +  q^. 

24.  ^x'  -  IZax^  +  13aV  -  ISa^a:  -  Sa^  by  2^2  -  3ax  -  a\ 

25.  4y^  -  26i/4  -  9y3  +  41y2  +  2«/  -  12  by  ^y'  +  2?/  -  3. 

26.  21a^  -  IQa^h  -  5a¥  +  IQa^^  +  2¥  by  3a^  -  ah  +  l)K 

27.  4rw^/i  -  4/?i3/i^  +  4m2n''  -  mn^  by  2//i-  -  2/?2/i  +  n-. 

28.  a^  +  6^  -  c^  +  3a6c  by  a  +  6  -  c. 

29.  ix2-tV^-^byix  +  i. 

30.  /^x^  +  f  1  by  f  a:  +  i 

31.  i^x^  -  ^\xhj  +  j\xy^  -^\  y^  by  ^x  -  ly. 

32.  ^m^  -  2m^  +  f fm^  +  f m  +  tV  by  f m^  -  f ?^i  -  i. 

33.  .16m%2  _  .Olps^^  by  Anfn  +  .lp^q\ 

34.  .04x3  _   i2y3  +  .i7a;^^2  _   i2xhj  by  .2a;  -  .Sy. 

35.  .002430;^  +  1  by  .3.r  +  1. 

36.  1  by  1  +  a;  to  five  terms  of  the  quotient. 

37.  1  by  1  -  a:  to  five  terms  of  the  quotient. 

38.  a:2»  +  2x''y"  +  if"  by  x"  +  y". 

39.  a'"  +  3a2"6"  +  3a"¥"  +  6^"  by  a"  +  6". 

40.  x^"+^  -  y^"+^  by  x"-^^  -  ?/"+^ 

41.  Show  that ^ ^—=  x  +  y  when  x  =  1,  and  y  =2. 

42.  Show  that  2m'*  —  bm^n  +  G/^i^/i^  -  Amm?  +  w*  divided 
by  m^  -  7nn  +  n-  equals  2m^  -  3mn  +  n^  when  wt  =  1,  and  n  =  l. 

43.  Show  that  a^  -  6^  divided  by  a  -  6  is  equal  to  a""  +  a^h 
+  a-6-  +  ab^  +  fe''  when  a  =  1,  and  fo  =  2. 

44.  The  area  of  a  rectangle  remains  1  while  the  base  and 
altitude  change.  Find  the  correspontling  values  of  the  base 
when  the  altitude  has  the  values  3,  2,  \,  .1,  .01,  .001,  .0001. 
Which  of  these  rectangles  has  the  least  perimeter?  Which 
has  the  greatest  perimeter? 

45.  In  the  fraction  -  let  x  take  on  the  values  1,  2,  4,  8,  IG, 
32,  64,  and  so  on.     So  far  as  possible,  represent  the  correspond- 


92  DIVISION  [Chap.  X. 

iiig  values  of  the  fraction  on  the  number  scale.  What  value 
is  approached  by  the  fraction  as  x  becomes  larger  ? 

46.   Find  the  corresponding  values  of  the  fraction 

when  X  takes  on  the  values  1,  2,  3,  4,  and  so  on.  So  far  as 
possible  represent  the  values  of  the  fraction  on  the  number 
scale.  Find  a  value  of  x  that  will  make  the  value  of  the  frac- 
tion greater  than  .999.  What  value  is  approached  by  the 
fraction  as  x  becomes  larger? 

64.    Literal  coefficients. 
Subtract : 
1.            ax  6.      av 

bx  -7v 


(a  -  b)x 

2.             cy 

7. 

-Qx 

-dy 

-mx 

(c  4-  d)y 

3.            ax 

8. 

rs 

-bx 

-s 

11. 

3x 

—nx 

12. 

ap  +  bq 
5p  -  mq 

13. 

kx'  -  9x 

14. 

nr-  -  mr 
-7r2  +  r 

15. 

ax  -  by  +  cz 
-2x  -dy  -z 

(a  +  b)x 

4.  am  9.  Sx'^y 

bm  ax'^y 

6.  3m  10.  ar 

am  br 

In  each  of  the  following  expressions  the  terms  have  a  com- 
mon factor.  Find  this  common  factor  and  write  each  ex- 
pression as  a  product. 

16.   ax  +  ay  -  az. 

Solution  :    The  common  factor  is  a.     Then, 

ax  +  ay  -  az  =  a{x  +  y  -  z). 


Art.  64]  EXERCISES  93 

17.  bx  +  by.  24.  a'-y  +  2aby  +  bhj. 

18.  6.r  -  by.  25.  x  -  3x  +  ax. 

19.  a2  -  3a.  26.  15a  -  20a6  +  5a. 

20.  p  +  prt.  27.  ahx-  +  a6?/  -  ab. 

21.  ?/  -  3?/.  28.  amx  +  a6w  -  ^amy. 

22.  7n  -  14//?  +  35fc.  29.  2a:  -  6?/  +  2. 

23.  /?/•  -  3/'-  +  br^.  30.  ax  -  bx  +  c.t  -  x. 

The  following  exercises  illustrate  the  use  of  the  division  of 
polj^nomials  in  the  solution  of  equations. 
Solve  and  check  results: 

31.  ax  -  a^  -  3a6  =  26^  -  bx. 
Solution  :     ax  -  a?  -  3a6  =  26-  -  bx. 

ax  +  hx  =  a?  -irZah  -{-  2W. 
(a  +  h)x  =  a2  +  3a6  +  262. 
a"  +  3a6  +  26^ 
a  +  6 
=  a  +  26. 

Check  :     Substituting  a  +  26  for  x  in  the  given  equation,  we  have, 
a{a  +  26)  -  a?  -  3a6  =  26^  -  b{a  +  26). 
Multiplying,         a^  +  2ab  -  a?  -  Sab  =  2¥  -  ab  -  2¥. 
Collecting  terms,  -ab  =  -ab. 

32.  2x  +  3ax  =  10a  +  15a2. 

33.  ax  +  bx  =  c{a  +  6). 

34.  ax  -  bx  =  a"  -  3a&  +  26-. 

35.  ax  +  bx  ==  c(a  +  b)(c  +  d). 

36.  ax  +  2ab  -  b^  =  a^-  +  bx. 

37.  ax  +  x  ^  a^  +  3a~  +  3a  +  1. 

38.  ax  -  bx  =  a^  -  b^. 

39.  mx  +  Sm  =  x  +  m-  +  2. 

40.  X  +  1  =  262  +  6  -  }j^^ 


94  DIVISION  [Chap.  X. 

REVIEW  EXERCISES  AND   PROBLEMS 

1.  Find  the  sums  of  (a)  Gx,  -2x,  -\x  ;  (b)  5{m+n),  -(m+n), 
-8(m+n);    (c)    7-3,3,   -11-3;    {d)    an,  3n,   -bn  ;    (e)    xy,  y,   -cy. 

2.  Arrange  Qx-  -  x  +  9x^  -  x*  +  5  according  to  ascending  powers 
of  X. 

3.  Arrange  a^  +  ia¥  +  a^b  -  a*¥  +  Qa-b-  according  to  descending 
powers  of  a  ;  of  6. 

4.  In  each  of  the  following  pairs  of  numbers  tell  which  number  is 
the  greater,  and  which  has  the  greater  absolute  value  :  (a)  -4,  7  ;  (6)  -9, 
-1 ;  (c)  -6, 0  ;  (d)  -5,  5. 

6.   State  a  rule  for  adding  like  terms. 

6.  State  a  rule  for  removing  parentheses;  (a)  when  preceded  by  the 
sign  +  ;   (b)  when  preceded  by  the  sign  -. 

7.  State  the  rule  for  combining  exponents  in  finding  the  product 
am  .  a»  •  a'". 

Complete  the  following  indicated  multiplications: 

8.  nv'-vi'-m.  12.    (f )- •  |.  16.   ^{2x-hj+\)- 

9.  2a^xybxyz\  13.    {-xY-{-x^).        17.   8xf  2x2  -  |  -  ^j. 

10.  {a^Y  ■  (aby.  14.   Sab  •  ia.  18.   3(2  •  3  +  4  •  3^  -  3^). 

a^ 

11.  {-a*y  ■  C2axy.        15.   nihi---^- 

19.  State  the  rule  of  exponents  in  dividing  am  by  a". 
Complete  the  following  indicated  divisions: 

20.  12a'xY  -  ^a'xif.  24.    (2a  +  26)  ^  2.         28.   ^-^jiy' 

21.  (23 .  4)  .  2.  25.    ^J!L±^m  .  29.  ^-±1^--^. 
^         ^                                        3m  -1 


22.    (23  +4)  ^2.  26. 


6  +  14.  {a-b)\ 

3  +  2  -(a  -  b) 


23.    (2a.  26).  2.  27.   ^-^^^  3^.  8a3  _  6a=  +  2a 


2a 


a^b  ■  ab'^ 


7x 


ab 
i.  Simplify :  X  -  3;/  -  (x  +  3?/)  -  { 1  +  [x  +  y  -  (3  +  2?/  -  x)  +  4] 


Art.  64]       REVIEW  EXERCISES  AND  PROBLEMS  95 

34.  Simplify  :  2a  +  (36  -  a  +  4)  -  [2  +  (36  -  4)  -  (1  +  «)]• 

35.  Multiply  500  +70+5  by  200  +30+4,  using  the  form  for 
multiplying  one  polynominal  by  another.  Compare  this  result  with  that 
obtained  by  multiplying  575  by  234  in  the  usual  way. 

36.  Compare  in  a  similar  way  the  products  (42 +|)  (18+1)  and 
42|  X  18i 

Solve  the  following  equations  for  x: 

Zl.   2x  -  4  +  3(x  -  1)  =  -2(x  -  2). 

38.  2x  -  a  =  3x  +  6, 

39.  ax  +  6^  =  6j;  +  a-. 

Af^    X       ,       3x      _     • 

40. 4  =  .^  +  7. 

a  la 

41.  mx  -  3/t  =  3»t  -  nx. 

42.  In  the  equation  2x  +  6  =  3,  give  6  numerical  values  such  that 
the  value  of  x  shall  be  (1)  a  positive  integer  ;  (2)  a  negative  integer  ; 
(3)  a  positive  fraction  ;  (4)  a  negative  fraction  ;  (5)  zero.  Which  of  these 
values  of  x  have  no  meaning  if  x  is  the  number  of  points  scored  in  a  foot- 
ball game? 

43.  Find  six  terms  in  the  quotient  1  -=-  (1  -  x).     What  is  the  difference 

between  the  sum  of  these  six  terms  and when  x  =  |?    When  x  =  .1? 

When  X  =  .01? 

44.  Show  that  the  numbers  of  the  series  .3,  .33,  .333,  .3333,  and  so  on 
approach  nearer  and  nearer  the  value  \. 

45.  If  A  is  the  area  of  a  square  whose  side  is  s,  then  A  =  s^.  By  what 
is  A  multipUed  if  s  is  multiplied  by  2?  If  s  is  multiplied  by  3?  By  4? 
By  10?     By  a? 

46.  If   t  =  10,    what  number  is   represented   by   Zt*  +  bt^  +  8<  +  5? 

47.  Write  86423  and  23090  in  terms  of  powers  of  t  =  10,  and  sub- 
tract 5  times  the  second  from  the  first.     Check  for  t  =  10. 

48.  Write  67731  and  211  in  terms  of  powers  of  t  =  10,  and  divide  the 
first  by  the  second.     Check  for  t  =  10. 


CHAPTER  XI 

LINEAR  EQUATIONS 

65.  Linear  equations.  A  great  many  of  the  equations 
we  have  solved  can  be  reduced  by  the  use  of  the  four  fundamental 
operations  to  the  form 

ax  +  h  =  0, 

where  x  represents  the  unknown,  and  a  and  b  are  numbers 
that  may  have  any  value  except  that  a  cannot  be  zero. 

Thus,  4x  +  3  =  X  -  7  can  be  reduced  to  3x  +  10  =  0,  which  is  of  the 
above  form,  where  a  =  3,  6  =  10.  Again,  ly  +  2  =3(1  +  ly)  can  be  put 
in  the  form  ay  +b  =  0,  where  a  =  ^,  6  =  -1 

Equations  which  can  be  reduced  to  the  form  ao:  +  &  =  0  by 
use  of  the  principles  given  in  Art.  38,  are  called  linear  equations 
in  one  unknown.  Such  equations  are  often  called  simple  equa- 
tions. 

EXERCISES  AND  PROBLEMS 

Write  the  following  equations  in  the  form  ax  +  b  ^  0  and 
point  out  the  values  of  a  and  b: 

1.  8:c  +  2  =  6x  +  6. 

2.  6a;  -  5  =  9a:  +  2. 

3.  5x  -3+  2\x  =  18  +  4a:. 

4.  3(a;  -  7)  =  a;  +  15. 

5.  81  -  4(a:  +  1)  =  a;  +  7. 

6.  25  -  6(a;  -  6)  =  20  -  (2a:  -  13). 

7.  13(2a:  -  1)  =  5(5a:  +  4). 
X      a:  +  4 


8. 


9  3 


.     7  -  3(a:  -  5)       , 
9.  ^  -  1. 

96 


Art.  65]  EXERCISES  97 

10.  ix-  l-U  =  U{Sx  +  l). 
Hint:    Multiply  each  member  by  36. 

Which  of  the  following  equations  can  be  reduced  to  the 
type  form  ax  +  b  =  0? 

11.  x{x  -  1)  =  4x  +  3. 

Solution:  Collect  terms  and  this  equation  reduces  to  a;^  -  5x  -  3  =  0. 
Since  it  contains  a  term  in  x^  it  is  not  of  the  form  ax  +b  =0. 

12.  (x+3)(x-2)  =4:{x-2). 

13.  x2  -  4a:  +  5  =  5  +  x\ 

14.  {2x  -  l)(3a:  +  1)  =  (Qx  -  12) (x  +  3). 

15.  {x  -  h){2x  -  9)  =  (a:  -  4)(2x  -  6). 

16.  x{x?  -  1)  =  24x  +  3. 

17.  (2x  -  l)(144a:  +  5)  -  26x  =  36x(8a;  +  1)  +  11. 

18.  (5  +  SY5  -  ^1  +  T  =  ^  +  12. 


2V      2     '  4 


19.  (a: -2)(a: +  2)  +1  =  ^^:^+30. 

20.  (a;  +  1)2  +  {x  -  2y  =  (x  -  l)(a;  +  5)  +  x\ 

Solve  the  following  equations  : 

21.  9a;  +  5a:  +  21  -  68  -  6  =  -2x  -  5. 

22.  3  •  5(a:  +  6)  +  5  •  7(1  +  2a-)  -  7  •  9(a:  -  8)  =  827. 


23. 

-  -  5a  =  X  -  23a. 
4 

24. 

X        XX        X 

4-5  +  3-6-^^- 

25. 

8a:-|=f^  +  153. 

26. 

<^-i)=<^3: 

27. 

a:      a:              a:      a: 
3"^2  +  ^"3~2' 

28. 

X  +  1      a:  +  2 
2             3 

98  LINEAR  EQUATIONS  [Chap.  XI. 

29.  I  +  3x  =  3  +  Ix. 

30.  4aj  +  5a  =  5(x  +  a). 

31.  (x  +  3)2  =  a;2  +  9. 

32.  (x  -  2)2  -  (x  -  3)2  =  X. 

33.  ^2  +  a;  +  1  =  (x  +  1)2. 

34.  x^  +  x  +  2  =  ix  +  2)2. 


35. 


?--  =  (^0' 


36.  (x  +  a){x  -  a)  =  x^  +  2x  +  2o?. 

37.  (X  +  l){^  +  2)  =  (X  +  2)(|  +  1 

38.  9x2  -  I  =  3(3a;2  ^  53,)^ 

39.  {2x  -  b){2x  +  5)  =  (4a;  -  ll)(a;  +  1). 

40.  When  4  is  subtracted  from  twice  a  number  the  result 
is  30.     What  is  the  number? 

41.  When  4  is  subtracted  from  a  number  and  the  result 
doubled  we  get  30.     What  is  the  number? 

42.  The  sum  of  two  consecutive  integers  is  27.  What  are 
the  numbers? 

43.  The  sum  of  three  consecutive  integers  is  27.  What 
are  the  numbers? 

44.  Is  it  possible  to  find  4  consecutive  integers  whose  sum 
is  27? 

45.  Find  4  consecutive  integers  whose  sum  is  46. 

46.  Find  three  consecutive  odd  integers  whose  sum  is  39. 

47.  Find  three  consecutive  even  integers  whose  sum  is  42. 

48.  A  teamster  contracts  to  haul  2100  bags  of  flour.  He 
makes  22  trips  with  his  dray  carrying  the  same  number  of 
bags  each  trip  except  the  last  one  when  he  carries  42  bags. 
How  many  bags  does  he  haul  each  of  the  first  22  trips? 

49.  The  perimeter  of  a  rectangle  whose  length  is  4  feet 
longer  than  its  width  is  28.     Find  the  dimensions. 

50.  The  length  of  a  rectangle  is  3  feet  more  than  twice  the 
width.     The  perimeter  is  42.     Find  the  dimensions. 


Art.  65]  PROBLEMS  99 

51.  A  cubic  foot  of  pure  water  weighs  62.5  pounds.  Twenty 
cubic  feet  of  water  weigh  15  pounds  less  than  22  cubic  feet 
of  ice.     What  is  the  weight  of  a  cubic  foot  of  ice? 

52.  One  pound  of  ice  occupies  30  cubic  inches  of  space. 
Five  pounds  of  ice  when  melted  decrease  12  cubic  inches  in 
volume.     What  volume  does  1  pound  of  water  occupy? 

53.  Eighty  gallons  of  water  from  the  Dead  Sea  weigh 
1  pound  less  than  100  gallons  of  pure  water.  The  weight  of  a 
gallon  of  pure  water  is  8.33  pounds.  What  is  the  weight  of  a 
gallon  of  water  from  the  Dead  Sea? 

54.  A  father  54  years  old  has  a  son  21  years  old.  How 
many  j^ears  ago  was  the  father  4  times  as  old  as  the  son? 

55.  What  number  is  to  be  subtracted  from  both  the  nu- 
merator and  denominator  of  ^f  in  order  that  the  new  fraction 
may  be  equal  to  ^? 

56.  The  denominator  of  a  fraction  is  12.  When  3  is  sub- 
tracted from  both  numerator  and  denominator  the  value  of 
the  fraction  is  decreased  by  ^\.  What  is  the  numerator  of 
the  first  fraction? 

57.  A  number  is  the  sum  of  two  parts.  The  first  is  3 
greater  than  half  the  number,  while  the  second  is  2  greater 
than  one^fourth  of  the  number.  What  is  the  number  and  how 
was  it  divided? 

58.  What  num})er  has  the  property  that  when  multiplied 
by  I  the  result  is  greater  by  one  than  when  multiplied  by  |? 


CHAPTER  XII 
IMPORTANT   TYPE  PRODUCTS 

66.  Certain  algebraic  products  occur  so  frequently  and 
are  so  useful  as  models  for  other  multiplications  that  they 
should  be  memorized. 

67.  Square  of  a  binominal.     Multiplying  a  +  6  by  a  +  6, 

we  find 

(a  +  by  =  a2  +  2ab  +  b^ 

which  may  be  translated  into  words  as  follows  : 

The  square  of  the  sum  of  two  numbers  is  the  square  of  the 
first,  plus  twice  their  product,  plus  the  square  of  the  second. 

In  a  similar  way,  we  find 

(a  _  6)2  =  «2  _  2ab  +  b^ 
or  in  words  : 

The  square  of  the  difference  of  two  numbers  is  the  square  of 
the  first,  minus  twice  their  product,  plus  the  square  of  the 
second. 

In  these  formulas,  a  and  b  represent  any  two  numbers,  or 
any  two  expressions. 

Example  1.     (10  +  5)^  =  10^  +  2  •  10  •  5  +  5^  =  225. 

Here  a  =  10,  and  6=5. 
Example  2.     [xif  +  (x  +  y)]^  =  (xif)^  +  2{xif){x  +  y)  +  {x  +  yY. 

Here  a  =  xy'^,  and  b  =  x  +  y. 

An  algebraic  expression  which  is  the  product  of  two  equal 
factors  is  called  a  perfect  square. 

100 


Art.  673 


EXERCISES 


101 


EXERCISES 

Square  the  following  binomials  by  the  above  rules  : 

1.  X  +  I/.  4.   X  +  c.  7.   .T  4-  2. 

2.  X  -y.  5.   2  -f  3.  8.   7  +  5. 

3.  n  -\-m.  6.   2  -  3.  9.   6  +  6. 

W 


a ^> 

F 

E 


-Ca+6)- 
FiG.  16 


Y 

.— (a-&9— 

X 

Fig.  17 


10.  Let  A  BCD  (Fig.  16)  be  a  square  whose  side  is  of  length 
a  +  b.  Its  area  is  then  (a  +  by.  Let  AEFG  be  a  square  of 
side  a.     From  the  figure  show  that  (a  +  h)-  =  a-  +  2ab  +  h-. 

11.  Let  TUVW  (Fig.  17)  be  a  square  whose  side  is  a.  Let 
XU  =  b.  Then  TXYZ  is  a  square  whose  side  is  (a  -  6). 
From  the  figure  show  (a  -  by  =  a?  -  2ab  +  b-. 

Square  the  following  numbers  by  expressing  each  number 
as  the  sum  or  difference  of  two  other  numbers  and  then  applying 
the  above  rules.     Work  each  exercise  in  two  different  ways. 

12.  39. 

Solution:  392  =  (30  +  9)2  =  30^  +  2  •  30  •  9  +  9^  =  1521. 
392  =  (40  -  1)2  =  402  -  2  •  40  •  1  +  12  .=  1521. 


16.  0.   17.  51.   18.  99. 


13.  41.   14.  69.   15. 

Square   the   following   according   to  the   aljove   rules   and 
verify  by  actual  multiplication  : 

19.  X-  -  y.  21.   xy  -  x.  23.   xy  +  2. 

20.  X  +  3y.  22.   ab  +  xy.  24.   2a  +  3a;. 


102  IMPORTANT  TYPE   PRODUCTS       [Chap.  XII. 

25.  xy  -  if.  28.    (a  +  h)  -f  c.  31.   x  +  y  -  z. 

26.  5fc  -  3/i.  29.   999.  32.   m  -  n  -  r. 

27.  a  +  116.  30.   x  ^- ij  +  z.  33.   2h  +  k  -  31. 

The  following  are  squares  of  binomials;  find  the  two  equal 
binomial  factors: 

34.  c2  +  2cd  +  d\  38.  4a'-  +  12ab  +  %^.  ' 

35.  a;2  -  2xy  +  if.  39.  9^2  +  62  +  1. 

36.  a;2  -  4a;  +  4.  40.  25a;2  +  20a;  +  4. 

37.  7/2  +  62/  +  9.  41.  49r2  -  42r  +  9. 

42.  Give  a  rule  for  finding  whether  or  not  a  trinomial  is  a 
perfect  square. 

Hint:  Two  terms  of  the  trinomial  must  be  perfect  squares.     What  is 
the  other  term? 

Some  of  the  following  are  perfect  squares;  find  them  and 
give  the  factors: 

43.  &  +  2cd  -  d\  48.   9  +  42  +  49. 

44.  a;2  -  lOx  +  25.  49.   4  -  5  +  25. 

45.  IQxY  -  8a;y  +  1.  50.   4^x''y*  +  Uxy"^  +  1. 

46.  9a2  +  7a  +  1.  51.   j  +  2a;  +  4. 

2a       1 

47.  xY~  +  xy  +  \.  52.   a2  +  --  +  -• 

68.    Product  of  the   sum  and  difference  of  two  numbers. 

By  carrying  out  the  multiplication,  we  find 
(^,  +  h)  {a  -  h)  =  (r~  -  h^'. 

In  words,  this  formula  reads  : 

The  'product  of  the  sum  and  difference  of  two  numbers  is  the 
difference  of  the  squares  of  the  two  numbers. 


AiiT.  68]  J<:XEHCISI':S  103 

EXERCISES 

Form  the  following  products  by  the  above  rule  and  verify 
by  actual  multiplication  : 

1.  {x  -  y)  {x  +  y).  7.  (3a  +  26)  (3a  -  26). 

2.  {c  +  d)  (c-d).  8.  (.T-7/)  (x  +  f-). 

3.  (x  +  2)  (x  -  2).  9.  (1  +  2)  (1  -  2). 

4.  (3.C  +  2j)  (3x  -  y).  10.  (10  -  1)  (10  +  1). 

5.  {x  -  2/2)  {x  +  1/).  11.  (a2  +  3)  (a2  -  3). 

6.  {x  +  2y)  (-  x  +  2y).  12.  {\  +  x)  {h  -  x). 

The  following  binomials  are  the  products  of  the  sum  and 
difference  of  the  same  two  numbers.     Find  the  numbers. 

13.  a?  -  W.  .  15.    16  -  z\  17.   ai"  -  6^. 

14.  4a;2  -  9?/.  16.     9  -  25.  18.   x^  -  y\ 

Perform  the  indicated  divisions  : 

19.  (a:2  _  y2)  ^  {x  +  y). 

20.  (a^  -  62)  -  {a}  -  6). 

21.  (s2  -  9r2)  -  (s  -  3r). 

22.  (16ri2  -  36m2)  -  (4w  -  6m). 

23.  (a2  +  2a6  +  6^  -  c^)  -  (a  +  6  +  c). 

24.  (x2  -  2xy  +  ?/2  -  92^)  ^  {x  -  y  ^  ?,z). 

25.  Give  a  rule  for  telling  whether  or  not  a  binomial 
is  the  product  of  the  sum  and  difference  of  the  same  two 
numbers. 

Which  of  the  following  are  the  products  of  a  sum  and  dif- 
ference?    Find  the  factors  of  those  which  are  such  products. 

26.  (a  -  6)2  -  c2.  29.   a2  +  62  -  c-. 

27.  a}  +  62.  30.   a2  +  a  +  1  -  62. 

28.  x^  +  2x2  +  1  -  1/^  31.   a2  +  2a  +  1  -  ¥. 


104  IMPORTANT  TYPE  PRODUCTS       [Chap.  XII. 

69.  Product  of  two  binomials  having  a  common  term.  Two 
binomials  having  a  common  term  can  be  written  in  the  form 
X  +  a  and  x  +  b.     By  actual  multiplication,  we  find 


x^  +  (a  +  h)x  +  ah 
or  (x  +  a)  {pp  +  6)  =  a-2  +  [a  +  b)x  +  ah. 

In  words  this  reads  : 

The  -product  of  two  binomials  having  a  common  term  equals 
the  square  of  the  common  term  plus  the  product  of  the  common 
term  by  the  sum  of  the  other  terms,  plus  the  product  of  the  other 
terms. 

EXERCISES 

Expand  by  the  above  formula  : 

1.  (x  +  2)  (x  +  d).  4.    (x-l)  {x-2). 

2.  (a  +  2)  (a  +  3).  5.    (a  -  4)  (a  +  5). 

3.  {x  +  5)  {x  +  2).  6.    (n  -  3)  (n  -  3). 

7.    (2a;  +  3)  {2x  +  1). 

Solution:    {2x  +  3)(2x  +  1)  =  {2x)^  +  (3  +  l)2x  +3-1 

8.  (3y  +  1)  (3y  +  2).  10.    (4  +  3a)  (4  -  2a). 

9.  (2a  +  b)  (2a  +  c).  11.    (ax  +  13)  (ax  -  1). 

The  following  trinomials  are  products  of  two  binomials 
having  a  common  term;  find  the  binomials: 

12.  x^  +  3x  +  2  =  (x  +  2)  ().  18.   y^  +  y  -  6. 

13.  x'-  +  4:X  +  3.  19.   a2  -  2a6  -  1562. 

14.  x"^  +  Qx  +  5.  20.   25  +  45  +  14. 

15.  n2  +  7n  +  10.  21.  u^  -  12u  +  32. 

16.  a2  -  6a  +  9.  22.    (x  +  1)2  +  3(rc  +  1)  -  4. 

17.  x^  -3x  +  2. 


Arts.  69,  70]      MISCELLANEOUS  EXERCISES  105 

MISCELLANEOUS  EXERCISES 

Form  the  following  products   according   to   the   foregoing 
rules  : 

1.  (^  -  xj  (^  +  xj.  9.    {ob  -  xu)  (ab  +  xy). 

2.  {x  +  1)  (x+  4).  10.    (x  +3)  (x-  16). 

3.  {x  +  3)  (x  +  3).  11.    (x2  +  f~)  (a;2  -  y^). 


'•(M)(M} 


12.    (2P  +  4)  (2P  +  6). 


5.  (xy  +  z)  (xy  -  z).  13.  (1  +  c)  (1  -  x). 

6.  (z  +  6)  (z  +  7).  14.  (9rH'-  +  1)  (9f-e~  -  1). 

7.  (a:  -I)  (x  +  2).  15.  [(a  +  6)  +  c]    [(a  +  b)  -  c]. 

8.  (a  -  4)  (a  +  4).  16.  (xY  +  7)  (xhf  -  2). 

The  following  expressions  are  either  the  products  of  the  sum 
and  difference  of  the  same  two  numbers,  or  products  of  two 
binomials  having  a  common  term ;  find  the  factors  of  each  : 

17.  x^  +  5x  +  4.  22.   x'-  -9x  +  20. 

18.  a"  +  7a  +  12.  23.   a^  +  2ab  +  b'- -  s^  +is  -  i. 

19.  100  -  9a\  24.   p^  +  7^^  +  iq^I 

20.  a2  +  4a  +  4.  25.    16  +  20.r  +  6^2. 

21.  (x2  -  2xy  +  y2)  _  z\ 

70.    Cube  of  a  binomiaL     By  actual  multiplication,  w^e  find 

(a  +  by  =  (a  +  6)  (a  +  b)  (a  +  b)  =  a  +  b 

a  +  b 


a" 

+  ab 
ab  +  ¥ 

a" 

+  2ab  +  62 
a  +  b 

a? 

+  2a-b  +  ab- 
a%  +  2a¥ 

+  63 

a?  +  3a26  +  3a62  +  6'. 


lOG  IMPORTANT  TYPE  PRODUCTS       [Chap.  XII. 

Tliat  is,  the  cube  of  the  sum  of  two  numbers  a  and  b  consists 

of  four  terms  as  follows  : 

The  first  term      =  a^      =  the  cube  of  the  first  number. 

The  second  term  =  da^b  =  three  times  the  square  of  the  first 

multiplied  by  the  second. 
The  third  term    =  Sab-  =  three  times  the  first  multiplied  by 

the  square  of  the  second. 
The  fourth  term  =  ¥      =  the  cube  of  the  second. 
As  a  formula,  we  have 

(a  +  by  =  a^  +  Sa^-b  +  3ab^  +  b\ 
In  a  similar  way,  we  find 

(a  -  by  =  a^  -  3a^h  +  3ab^  -  bK 

EXERCISES 

Expand  the  following  by  the  foregoing  rule  or  formula  : 

1.  (c  +  (ly.  4.   (x  +  ly.  7.   (3a  -  by. 

2.  {x  -  yy.  5.    (z  +  Sy.  8.    (5.t  -  2)^ 

3.  (a +  2)3.  6.    (2x  +  yy.  9.    {xy  +  ly. 

10.    {xy  +  2zy. 

71.    Square  of  a  trinomial.     By  actual  multiplication,  we 
find 

(^a  -\-b  +  cy  =  a-  +  b^  +  c"^  +  2ab  +  2ac  +  2bc. 

That  is,  the  square  of  a  trinomial  equals  the  sum  of  the  squares 
of  its  terms  plus  twice  the  product  of  each  term  by  each  succeeding 
term. 

EXERCISES 

1.  By  actual  multiphcation  prove  : 

(a  +  fe  -  c)2  =  a2  +  ?;2  +  c2  +  2ab  -  2ac  -  2bc. 

2.  Prove  {a  -  b  -  c)-  =  a~  +  ¥  +  c^  -  2ah  -  2ac  +  26c. 

3.  Prove    (a  -  6  +  c)^  =  a^  +  6^  ^  c^  -  2ab  +  2ac  -  26c. 


Art.  713 


EXERCISES 


107 


the 
the 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 


10. 
11. 
12. 
13. 


By  use  of  the  foregoing  formuhis  exi)aiKl  : 

4.  {x  +  y+zy.  8.    (1  +2  +  3)2. 

5.  (x-y  +  ly.  9.    (1  +  1  +  1)\ 

6.  (a  +  26  +  3c)2.  10.    [(a  +  6)  +  .r  +  y]\ 

7.  (2x-y  +  Szy.  11.     {a -2b-  Sc^. 

MISCELLANEOUS  EXERCISES 

Perform  the  following  multiplications  and  divisions,  using 
type  products  of  this  chapter  whenever  possible.  Check 
results  by  substituting  special  numerical  values. 


(1  +  ay-. 

(1  +  70  (1  +  2/1). 
(1  -  n)  (1  +  2n). 
(1  -  n)  (1  -  2n). 
(1  +  n)  (1  -  2n). 
(1  -  3fl)  (1  +  3a). 
(1  +  2)1 

(1  +  ay. 

(2x  -  3)2. 

(1  -  X  -  y)  (1  -  .r  +  y). 

(xy  -  8)  {xy  +  1). 

(x  +  y  -  8)  {x  +  y  +  1). 

(1  +  2)^ 


14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 

22. 

23. 
24. 
25. 


(1  +  a  +  2a)2. 
(4a-2  +  4.T  +  1)  -h  {2x  +  1). 
(a'2  _  3x  +  2)  -  {x  -  2). 
(a  -  I)  (a  +  i). 
(a  +  i)  (a  -  ^). 
(a  +  !)(«  +  ^). 
(a  -  i)  (a  -  i). 
(101)2. 
4  -  16c2 
2  -  4c  ■ 


(101)  (99). 

(.t2  +  2Qx  +  169) 


(a:  +  13). 


(a2  +  ax  -  2x2)  h-  (a  +  2x). 


^~b> 


I 

a; 


Fia.  18 


x-*< a 

Fig.  19 


26.   What  are  the  dimensions  of  the  rectangles  in  Fig.  18? 
Show  that  the  two  rectangles  can  be  joined  together  so  as  to 


108  IMPORTANT  TYPE  PRODUCTS       [Chap.  XII. 

make  a  rectangle  whose  length  is  a  +  6,  and  whose  width  is 
a  -  b.     From  the  figure  show  that  (a  +  b)  {a  -  b)  =  a^  -  ¥. 

27.  Fig.  19  is  a  rectangle  of  width  x  -\-  a  and  height  x  -\-b. 
From  the  figure  show  that  {x  -\-  a)  {x  -{■  b)  =  x"^  +  {a  -{-  b)x  +  ah. 

Fill  out  the  blanks  in  the  following  : 

28.  a?b^  -  Mc'd?  =  {ab  +  8cd)(  ). 

29.  a^  +  'ia?x  +  Sax^  +  x^  =  (a^  +  2ax  +  x'^){  ). 

30.  4  +  2x  +  2?/  +  a:;y  =  (2  +  x)(  ). 

31.  {xY-  -  2xy  +  1)  =  {xij  -  1)(  ). 

32.  (2x  +  a)  (2x  +  6)  =  4a;2  +  (  )  +  ab. 

Remove  parentheses  and  unite  terms  where  possible  : 

33.  (2a;  -  3)2  +  3  (3a:  -  2)2. 

34.  (xy  -  zY  +  {xy  +  z)-. 

35.  6a:(a  -  1)  +  2(3x  -  a^. 

36.  6x(a  -  1)  +  2(a  -  3x)2. 

37.  a{\)-a)  -  (6^  -  a^). 

38.  (a2  -  62)  -  (a  -  b)2. 

39.  6(22  _  32)  _  12(2  -  3)2. 

40.  (5»  -  33)  -  (5  -  3)^ 

41.  {0?  -  ¥)  -  (a  -  by. 

42.  Show  by  multiplying  that 

a(a  -  x)(y  +  z)  +  x(a  -  y){a  -  z)  -  2ayz 
=  a(a  -  y)(x  +  z)  -  y{x  -  a){a  -  z)  -  2axz. 


CHAPTER  XIII 
FACTORING 

73.  Prime  factors  in  arithmetic.  One  of  the  pro})loms  of 
arithmetic  is  that  of  finding  the  factors  of  a  given  number.  A 
factor  is  one  of  two  or  more  numbers  whose  product  is  the  given 
number.  Thus,  30  may  be  written  2-15,  5-6,  3-10,  or  2  •  3  •  5. 
The  numbers,  2,  3,  5,  6,  10,  and  15  are  all  integral  factors  of 
30.  If  we  consider  fractions  we  may  go  on  indefinitely,  for 
we  may  write  30  =  |  •  24,  30  =  |  •  |  •  36,  30  =  |  •  Y  '  Vt',  and 
so  on. 

The  important  problem  in  this  connection,  however,  is  find- 
ing the  prime  factors.  A  prime  factor  is  an  integral  factor  which 
is  the  product  of  no  two  integers  except  itself  and  unity.  There 
are  many  sets  of  numbers  whose  product  is  30,  but  there  is 
only  one  set  of  prime  factors,  that  is,  2,  3,  and  5.  Any  integer 
has  only  one  set  of  prime  factors. 

By  the  factors  of  a  number  we  shall  often  mean  its  prime 
factors  although  the  word  prime  is  omitted. 

EXERCISES 

Separate  the  following  numbers  into  products  of  prime 
factors;  find  for  each  number  two  sets  of  factors  which  are  not 
prime : 

1.  12.  3.     30.  5.   32.  7.     99.  9.    170. 

2.  15.  4.    120.  6.   81.  8.    100. 

73.    Prime  factors  in  algebra.     The  expressions  which  we 

propose  to  treat  in  this  chapter  involve  a  definite  number  of 

additions,  subtractions,  multiplications,  and  divisions,  but  no 

other  operations.      Such  expressions  are  said   to  be  rational 

109 


no  FACTORING  [Chap.  XIII. 

expressions.  A  factor  of  a  rational  expression  is  one  of  two 
or  more  rational  expressions  whose  product  is  the  given  ex- 
pression. 

An  expression  that  is  rational  with  respect  to  a  given  letter 
contains  no  indicated  root  of  that  letter. 

Thus,  a,  -  +  -,  a^  +  — -  are  all  rational  with  respect  to  a  and  x.     The 
'     '  a;      a  3a 

expression  a  -V  ax  ^  Vx  is  rational  with  respect  to  a,  but  not  with  respect 

to  x.     The  expression   involves,  besides   the  operations  of  addition  and 

multiplication,  the  operation  of  extracting  the  square  root. 

An  expression  is  integral  with  respect  to  a  letter  if  this  letter 
does  not  occur  in  any  denominator. 

Thus,  a^  +  ^j—  is  integral  with  respect  to  x,  but  fractional  with  respect 
Sa 
to  a. 

A  factor  is  rational  and  integral  if  it  is  rational  and  integral 
with  respect  to  all  the  letters  contained  in  it. 

Thus,  x2  +  xy  +  5a^  is  rational  and  integral. 

In  this  chapter  the  word  factor  is  to  mean  rational  and 
integral  factor. 

Thus,  the  factors  of  x^  -  if  are  x  -  y  and  x  +y.  Although  x'^  -  ?/ 
=  Vx^  -  y^  •  Vx^  -  y^,  we  shall  not  consider  Vx^  -  y-  as  a  factor  of  x^  -  if. 
Again,  the  factors  of  j  -  x-  are  \  -  x  and  \  +  x  ;  the  factors  of  2x^  +  2x'^  + 
2x  are  2,  x,  and  x^  +  x  +  1. 

A  factor  is  said  to  be  prime  if  it  contains  no  factor  except 
itself  and  one.  The  different  prime  factors  of  Iba^h^c  are  3,  5, 
a,  h,  and  c.  15,  a^,  h^,  ab  are  also  factors,  but  not  prime  factors. 
In  algebra,  as  in  arithmetic,  the  most  important  problem  of 
factoring  is  that  of  finding  the  prime  factors. 

74.  Factors  of  monomials.  The  factors  of  a  monomial 
are  evident  by  inspection. 

Thus,  the  different  prime  factors  of  21aV  are  3,  7,  a,  and  x. 


Arts.  74,  75]  JMONOAIIAL  FACTORS  111 

EXERCISES 

Name  the  different  prime  factors  of-  the  following  : 

1.  Ga-b\  4.   91ab-(^d\  7.   a^¥c^d\ 

2.  12xYz.  5.   32pV-  8.    imx^z\ 

3.  42mhi'xhj.  6.   33  •  2'sH'.  9.    1728{abcy. 

10.    -lOSixi/z'y. 

11.  If  ab-  is  considered  as  one  factor  of  Qa~b-,  what  is  the 
other? 

12.  If  42m^n^x^y  is  considered  as  the  product  of  two  factors 
and  one  factor  is  6mx,  what  is  the  other? 

13.  ^2p-q^r  is  the  product  of  three  factors.  Two  of  them 
are  2p  and  pqr.     What  is  the  other  factor? 

14.  If  a^b^(?d''  is  obtained  by  multiplying  together  three 
expressions,  two  of  which  are  a^c-  and  bd^,  what  is  the  third? 

15.  The  expression  'dXaH^&d^  is  obtained  by  multiplying 
together  three  expressions.  One  of  these  is  VZbcd.  Write  down 
three  possible  pairs  of  the  other  two  factors. 

75.  Monomial  factors  in  polynomials.  If  each  term  of  a 
polj^nomial  contains  the  same  monomial  factor,  then  this 
monomial  factor  is  a  factor  of  the  polynomial,  and  the  problem 
is  to  obtain  the  other  factor.  (See  Arts.  45,  53,  62.)  A  type 
form  of  expression  coming  under  this  class  is 

ax  -\-  ay  -  az  =  a{x  +  y  -  z), 

and  the  factors  of  ax  +  ay  -  az  are  a  and  x  +  y  -  z. 

Thus,  in  factoring  60*6  -  12a^6'*  -  Sa^b-  we  find  that  each 
term   of  the   polynomial   contains   Sa^b.     Then 

Qa'b  -  12a'b*  -  Za'b^-  =  2,a^b{2a  -  W  -  b). 

EXERCISES 

Factor  the  following  : 

1.  mx  -  my.  3.   x-  +  x. 

2.  ix  -  8.  4.   x'-y  -  xy. 


112  FACTORING  [Chap.  XIII. 

5.  6a2  +  16a.  8.   2x  +  Ay  -  8z. 

6.  3a;  +  9xy  +  lSx\  9.   Qa^  -  9a^  +  da. 

7.  a^  +  a?h  +  a"x.  10.   5x  +  20a;2  +  15x1 

11.  2a;  +  2|/  +  ax  +  ay  =  2(a;  +  y)  +  a{x  +  y)  =  (2  +  a)(  ). 

12.  4aa;  +  46x  -  3a  -  36  +  5ay  +  bhy 

=  4a:(a  +  6)  -  3(a  +  6)  +  5i/(a+  6)  =  (  )(  )^ 

13.  3a  +  36  +  3c  -  2a.T  -  26x  -  2ca; 

=  3(a  +  6  +  c)  -  2x  (a  +  6  +  c)  =  (  )(  ). 

14.  Write  down  a  rule  for  factoring  polynomials  having  a 
common  monomial  factor. 

76.  Factors  found  by  grouping  terms.  As  shown  in  Exer- 
cises 11,  12  and  13  of  the  last  article,  a  polynomial  may  have 
polynomial  factors  which  may  be  found  by  grouping  the  terms 
properly.  A  typical  example  of  this  class  is  ax  +  ay  +  hx  -\-  by. 
Collecting  the  terms  in  parentheses,  we  get 

(ax  +  ay)  +  (hx  +  by)  =  a{x  +  y)  +  6(a;  +  y). 

Each  group  of  terms  contains  x  +  y,  which  then  may  be  factored 
out.     This  gives 

a(a;  +  y)  +  6(a:  +  y)  =  (x  +  y)(a  +  6). 
Hence, 

ax  +  mj  +  6.r  +  hy  =  (a?  +  y){(i  +  h). 

EXERCISES 

Factor  the  following  : 

1.  2(a  +  6)  +  x{a  +  6).  3.   2(2a:  +  y)  +  a;(2rc  +  y). 

2.  a{x  +  y)  +  (x  +  y).  4.   a(6  -  c)  +  6(6  -  c). 

5.  Zx{2y  -  30)  -  4y(2y  -  3^). 

6.  2(4a  -  3c)  +  3(4a  -  3c). 

7.  2(a  +  6  +  c)  +  a(a  +  6  +  c). 

8.  x{x  +  y)  +  y{x  +  y)  +  2(.r  +  y). 


Akts.  7G,  77]        DIFFERENCE  OF  SQUARES  113 

9.  a{a  +  b)  +  2a{a  +  &)  +  h{a  +  h). 
10.   nix  —  y)  -\-  m{y  —  x). 
Hint:     Write  the  expression  n{x  -  y)  -  m{x  -  y). 

MISCELLANEOUS  EXERCISES 
Factor  the  following : 

1.  -  3a(6  +  c)  -  4(6  +  c)  +  2(6  +  c). 

2.  a(a  +  6  +  c)  +  6(a  +  6  +  c). 

3.  4a26  -  106c  +  6a6. 

4.  8^2  +  4a:3  -  2x'  +  Gx^. 

5.  a(a  -  6)  +  2(6  -  a). 

6.  a%  -  a^¥  +  0^63  -  ah\ 

7.  a(a  +  6  +  c)  +  6(a  +  6  +  c)  +  c(a  +  6  +  c). 

8.  ^x^y  -  12a;Y  -  l%xY  +  8a;^y^ 

9.  Write  the  prime  factors  of  -&2bx'^y^. 
10.   Write  the  prime  factors  of  27xy(x  +  yy\ 

77.   Difference  of  two  squares.     The  typical  form  for  this 

case  is 

a2  -  62, 

which  we  have  seen  to  be  the  product  of  the  sum  and  difference 
of  a  and  6.     Hence, 

a2  _  j[>2  =  (^a  _|_  ft)(^,  _  ft). 

EXERCISES 

8.  x'  -  16. 

9.  X*  -  1. 

10.  16a4  -  81. 

11.  (2a  -  by  -  h\ 

12.  25  -  4(a;  -  y)^. 

13.  (a  +  6)2  -  (.r  +  y)\ 
7.   .T^  -  y\                              14.  (2a  -  6)2  -  9(x  -  1)2. 

Solution:   {x'  -  j/^)  =  (x^  +  y-'){x'  -  y^)  =  (x^  +  y^)(x  +y){x  -  y). 


Factor  the  following 

1. 

a;'^ 

-z\ 

2. 

a;2 

-  1. 

3. 

a2 

-4. 

4. 

n2 

-9. 

5. 

1  - 

-a2. 

6. 

9  - 

-4. 

114  FACTORING  [Chap.  XIII. 

MISCELLANEOUS  EXERCISES 
Factor  : 

1.   3abc  +  9arbc.  "      2.   a^x*  -  1. 

3.  pip  -  q)  -  q{q  -  p)  +  2{q  -  p). 

4.  a^x^  -  hY-  10.   x^  -  y\ 

5.  x'^  -  y^.  11.   ab  +  bx  +  ac  +  ex. 

6.  36x«  -  1.  12.   26  -  1. 

7.  2x1/2  -  Sx^yz  -  IQxyh.  13.    17a¥xy^  -  Sia^b^xHf. 

8.  WLT  +  7ny  +  nx  +  ny.  14.    169x^  -  ?/^. 

9.  a^  -  ¥.  15.   a^x-  -  a. 

78.   Trinomial  squares.     We  have  seen  that 

a'  +  2ab  +  62  =  (a  .+  by, 
and  a2  -  2a6  +  b^  =  (a  -  by, 

from  which  we  see  that  if  in  a  trinomial  the  middle  term  is 
twice  the  product  of  the  square  roots  of  the  other  two  terms, 
then  the  trinomial  is  a  perfect  square;  that  is,  it  is  a  product 
of  two  equal  factors. 

Thus,  in  x-y^  -  2xy  +  1 ,  we  have  2xy  =  2  •  y/x^  •  a/T  and 
the  trinomial  is  a  perfect  square,  [xy  -  1)". 

EXERCISES 

Test  the  following  trinomials  to  see  if  they  are  perfect 
squares.     If  they  satisfy  the  tests,  find  the  two  equal  factors. 

1.  a;2  +  2xy  +  y"^.  5.  a^  +  2a  +  1. 

2.  x"^  -  2xy  +  y2.  6.  a;2  +  2>xy  +  y'^. 

3.  x^  -  xy  +  y^.  7.  4x^  +  4xy  +  2y\ 

4.  a;2  +  2xy  +  y\  8.  4  +  16  +  16. 

9.   702 +2 -70 -6+62. 


Art.  78]  INIISCELLANEOUS  EXERCISES  115 

111  each  of  tlie  following  expressions  replace  the  parentheses 
by  a  term  which  will  make  the  trinomial  a  perfect  square: 

10.  .T-  +  (  )  +  if.  13.   400  +  (  )  +  32. 

11.  a'-  +  {  )  +  1.  14.  rc2  +  4.T?/  +  (  ). 

12.  4x2  +  (  )  +  fl2^  15,   25?/  +  (  )  +  4. 

16.    lG9/r  +  (  )  +  imnC-. 

17.  The  middle  term  of  a  trinomial  is  2.ryz.  Find  three 
possible  pairs  of  terms  which  will  make  the  trinomial  a  perfect 
square. 

Each  of  the  following  terms  may  be  considered  as  the  niidiUe 
term  of  many  trinomial  squares.  Find  three  such  trinomial 
squares  for  each  exercise. 

18.  2abc.  19.   Qxyz.  20.   4a^b. 
Separate  each  of  the  following  into  two  factors  : 

21.  225xhf  +  120x''2/*  +  16xY. 

22.  36a262  +  QOaW  +  25a*b\ 

23.  9a8  +  42a7&  +  49a«62. 


MISCELLANEOUS  EXERCISES 

Factor  if  possible  : 

1. 

{x  -  vY  -  z\ 

2. 

-ab-cW  +  2,a'bHH  +  ^a}b"-cH\ 

Find  two  factors  only. 

3. 

ax  -  bx  +  ay  -  by. 

4. 

4a2  -  12a  +  9. 

5. 

a262  +  4^ab  +  4. 

6. 

a262  +  2abx  +  x^. 

7. 

a2"  -  1. 

8. 

0252  4-  lOabxy  +  25zV. 

9. 

x^  +  xy  +  2x  +  2y. 

10. 

x^  -  6a;2  +  9. 

11. 

3a(a  +  6  +  c)  -  (a  +  6  +  c). 

12. 

x^  -  2x?  +  1. 

116  FACTORING  [Chap.  XIII. 

In  each  of  the  following  replace  the  parentheses  by  a  term 
which  will  make  the  trinomial  a  perfect  square: 

13.  xY  +  {)  +  z\  14.   4a2  +  4a  +  (  ). 

15.  (  )  +  42ah  +  9. 

16.  Find  three  trinomials  which  are  perfect  squares  and 
which  have  2x-y  for  a  middle  term. 

17.  Factor  a^  +  2axHj  +  x^- 

18.  Factor  .t^"  -  if. 

19.  Factor  2ax  +  2aij  -  36a;  -  Zhij. 

20.  Factor  \mx^  -  \d,2x'^y  +  492/2. 

79.  Trinomials  of  the  form  x^  -\-  {a  -\-  b)x  +  ah.  We  have 
found  that  the  product  of  two  binomials  having  a  common 
term  is  given  by  the  formula 

{x  +  o)  {x  +  h)  =  x"^  -{-  {a  -\-  h)x  +  ah. 

Here  it  is  to  be  noted  that  the  coefficient  of  x  is  the  algebraic  sum  of 
a  and  h  and  that  the  last  term  is  their  product.     For  example, 

(x  +4)(x  +3)  =x2  +7a;  +  12, 

(x  -4)(x  -3)  =x2  -7x  +12, 

(x  -4)(x  +3)  =.c2  -X  -    12, 

(a;  +4)(x  -3)  =x2  +x  -    12. 

If  a  trinomial  comes  under  this  type,  it  is  possible  to  find  two 
numbers  whose  sum  is  the  coefficient  of  x  and  whose  product 
is  the  last  term.  These  trinomials  can  usually  be  factored 
by  inspection. 

EXERCISES 
From  the  following,  pick  out  those  trinomials  which  are  of 
the  form  x^  +-  (a  +  h)x  +  ah  and  find  the  factors. 

1.   .T"  -  5x  +  6. 

Solution:  We  are  to  find  two  numbers  whose  sum  is  -5  and  whose 
product  is  6.  Such  numbers  are  -2  and  -3.  Hence,  x^  -  5x  +  6  = 
(x-2)(x  -3). 


Art.  79]  EXERCISES  117 

2.  .r^  +  5.r  +  6. 

3.  X-  -  X  -  (). 

Solution:  We    are   to   find    two   numbers  whose  sum  is    -1    and 
whose  product  is  -6.     Such  numbers  are  2  and  -3.     Hence,  x'^  -  x  -  6  = 

(x+2)(x-3). 

4.  X-  +  X  -  Q. 

5.  a;2  +  8x  +  2. 

Solution  :  It  is  impossible  to  find  two  integers  whose  product  is  2  and 
whose  sum  is  8. 

6.  x^  +  7x  +  2.         9.   X-  -  X  -  2.  12.   a-  +  5a  +  1. 

7.  a;2  +  3x  +  2.       10.   x^~  +  x  -  2.  13.   a-'  +  2a  +  1. 

8.  x^  -Sx  +  2.       11.   a2  +  4a  +  1.  14.   7/  +  Qij  +  8. 

MISCELLANEOUS  EXERCISES 
Factor  : 

1.  4x{a  -h)  -  3(6  -  a)  +  5y{a  -  b). 

2.  z-  -  Qz  +  8.  6.   ax^  -  Sax  +  15a. 

3.  x'-  -2x  -  8.  7.   n2  +  72.  -  12. 

4.  62  +  26  -  8.  8.   x^  -  7f. 

5.  6xy  +  46?/  -  cxy  -  4c?/.  9.   p-  +  7p  +  3. 

10.  a;2  +  4x  +  3. 

11.  Fill   out   the   parentheses  to  make    (  )  +  4.r?/  +  y-   a 
perfect  square. 

12.  Fill  out  the  parentheses  to  make  81a-.r-  +  (  )  +  4a^  a 
perfect  square. 

Factor : 

13.  5  +  6a:  +  x^.  16.  ax-  +  9ax  +  20a. 

14.  a"  +  12a;  -  28.  17.  9a-  -  6^ 

15.  2x'  -x^+4x-2.  18.  ?/  -4y  -21. 


118  FACTORING  [Chap.  XIII. 

80.    General  quadratic  trinomial,  ax-  +  hx  +  c.    If  we  mul- 
tiply together  the  two  binomials  2>x  +  5  and  2x  +  3,  we  find: 


3a:  +5 

3a;     5 

2a:  +3 

X 

6x2  +  iO:c 

2x    3 

9a; +  15 

^x'  +  19x  +  15 

two  products  2x 

5  = 

=  10a; 

and  3a;  •  3  =  9a;  are  called 

cross  products. 

The  product  6a;-  +  19a;  +  15  is  a  trinomial  of  the  form 

ax^  -{-hx  -\-  c. 

While  the  product  of  two  binomials  like  3a;  +  5  and  2a;  +  3 
is  a  trinomial  of  this  form,  yet  all  trinomials  of  the  type 
ax"'  +  hx  +  c  cannot  be  factored  into  such  binomial  factors. 
If  the  trinomial  can  be  factored  it  is  often  easily  done  by 
inspection. 

Example.     Factor  Gx^  +  19x  +  15. 

The  product  of  the  first  terms  of  the  binomial  factors  is  6a;^.  The 
first  terms  are  then  2x  and  Sx,  or  6x  and  x,  if  the  coefficients  are  integers. 

The  product  of  the  second  terms  of  the  binomial  factors  is  15.  The 
second  terms  are  then  ±  3  and  ±  5  or  ±1  and  ±  15,  if  the  second  terms  are 
integers.  Since  the  middle  term  of  the  trinomial  is  positive  we  keep  only 
the  positive  terms  3  and  5,  and  1  and  15. 

We  have  now  to  pick  out  two  binomials  having  2x  and  3x,  or  6x  and 
X  for  first  terms  and  with  3  and  5  or  15  and  1  for  second  terms  in  such  a 
way  that  the  middle  term  of  the  product  is  19.c.  That  is,  the  binomials 
are  chosen  so  that  the  algebraic  sum  of  the  cross  products  is  19.i'.  By 
trial,  we  find  the  factors  to  be  3.r  +  5  and  2x  +  3. 

EXERCISES 

Find  the  following  products  : 

1.  (2.T  +  3)  (5.r  +  4).  5.  (2a  +  1)  (3a  +  1). 

2.  (2.r  -  3)  (5.r  +  4).  6.  (2c  -  4)  (2c  +  3). 

3.  (2.T  -  3)  (5.1;  -  4).  7.  (2.r  +  5)  (2.r  +  5). 

4.  {2x  +  3)  (.5.r  -  4).  8.  (5//  +  4)  {2ij  +  3). 


Arts.  80,  81]  EXERCISES  ll'J 

Factor  the  following  : 

9.  2x-'  +  Ix  +  3.  15.  6x2  ^  7^  _^  2. 

10.  3.1--  +  5.C  +  2.  16.  6x2  _  8.r  +  2. 

11.  5x2  +  -jy.  ^  2.  17.  6x2  _  7^^.  ^  2. 

12.  5x2  _  ly.  _,_  2.  18.  10x2  +  llx  +  3. 

13.  2x2  -  5x  +  3.  19.  10x2  -  13x  +  3. 

14.  2x2  -  7x  +  5.  20.  15x2  +  13x  +  2. 

21.  Show  that  the  trinomial  x2  +  (a  +  h)x  +  a&  is  a  special 
case  under  the  type  ox2  -\-hx  -\-  c. 

Factor  : 

22.  x2  +  9x  +  14.  24.   x2  -  7x  -  18. 

23.  x2  -  4x  -  21.  25.   x2  -  15x  -  34. 

MISCELLANEOUS  EXERCISES 

1.  2(x  +  y)  +  a(x  +  y).  3.   x^  -  x2  -  x  +  1. 

2.  9(x  +  yf  -  4.  4.   8x2  +  22x  +  15. 

5.  8x2  +  23x  +  15. 

6.  4x(a  +  6)  -  3(a  +  6)  +  by{a  +  h). 

7.  p2  +  4p  _  21.  14.  8x2  _  23a;  +  15. 

8.  a-  +  9a6  +  862.  15.   f  j^  {x -\-  y)t  +  xy. 

9.  a?¥  -  a}¥  -ah  +  \.  16.   Sx^  -  6ax2  +  x  -  2a. 

10.  ax  +  hy  -ay  -  hx.  17.   8x2  _  i21x  +  15. 

11.  (a  +  hy  -  c\  18.   8.r2  +  43x  +  15. 

12.  (2x  +  ZyY  -  1.  19.    (rt  +  6  +  c)2  -  d\ 

13.  8x2  ^  i21x  +  15.  20.   4a.r'  +  8flx  -  8a  -  4a.r2 

81.    Sum  and  difference  of  two  cubes.     By  actual  multipli- 
cation, we  find 

a^  +  63  =  {a  +  h){a~  -  ah  +  b^), 
and  (i^  -  h^  =  {(t  -  h){a-  +  r/6  +  b"^). 

Any  expression  which  may  be  written  as  the  sum  or  difference 
of  two  cubes  can  be  considered  as  the  i)roduct  of  a  binomial 
factor  and  a  trinomial  factor. 


120  FACTORING  [Chap.  XIII. 

EXERCISES 

Factor  the  following  : 
1.   x"  +  27. 

Solution  :  We  may  write  the  expression  in  the  form  x^  +  3^  from 
which  we  get  x'  +  3^  =  (x  +  3)  (x^  -  3x  +  3^), 

x^  +27    =  (x  +3)(x2  -3x  +9). 

The  factor  x~  -  3x  +  9  cannot  be  factored  since  there  are  no  two  integers 
whose  product  is  9  and  whose  sum  is  -3. 

6.  8-1. 

7.  8a;3  -  125. 

8.  if  -  125^3^ 

Find  two  factors  only.     Hint:  ¥=  {b'^y. 
Find  two  factors  only. 

11.  x^  -  y^.    Find  two  factors  only. 

12.  x^  -  y^. 

Hint:  This  expression  can  be  written  as  the  difference  of  two  cubes 
or  the  difference  of  two  squares.  Factor  by  both  methods.  Which  is 
the  easier? 


2. 

x'  -  27. 

3. 

a'  -  1. 

4. 

a'  +  l. 

5. 

8  +  1. 

9. 

a'  +  ¥. 

0. 

x^  -  y^. 

13. 

64x«  -  y\ 

14. 

a^  +  h^.    Find  two  factors  only. 

15. 

125x^1/3  +  82^ 

16. 

a.12  _(_  ^12     jTjnd  two  factors  only. 

17. 

^12  _  ^6     pjj^(j  two  factors  only. 

18. 

x^"  -  I.    Find  two  factors  only. 

19. 

^sn  ^  ^3n     Yind  two  factors  only. 

20. 

(a  +  hy  +  (a-  by. 

21. 

(a  -  xy  -  x\ 

22. 

(a  -  xy  +  x\ 

23. 

(^2  -  1)3  +  (^2  +  1)3.    Find  four  factors 

24. 

125(x  +  yy  +  8^3. 

25. 

1000  -  1. 

Arts.  81,  82]         SUMMARY  OF  FACTORING  121 

26.  Give  in  words  a  rule  for  factoring  the  sum  of  two  cubes. 

27.  Give  a  rule  for  factoring  the  difference  of  two  cubes. 

82.  Summary  of  factoring.  No  simple  general  rules  for 
factoring  can  be  given,  but  a  few  suggestions  will  be  helpful. 

(1)  First  take  out  all  monomial  factors,  not  forgetting  factors 
expressed  in  Arabic  numerals. 

(2)  After  the  monomial  factors,  if  any,  have  been  re- 
moved, the  number  of  terms  will  usually  be  the  best  guide  in 
factoring  further. 

(a)  Binomials  are  factored  as  — 

The  difference  of  two  squares,  a^  -  ¥. 
The  difference  of  two  cubes,  a'  -  ¥. 
The  sum  of  two  cubes,  a^  +  ¥. 

(b)  Trinomials  are  factored  — 

As  trinomial  squares,  a^  +  2ab  +  b~. 

By  inspection,  x~  +  (a  +  h)x  +  ah,  and  ax^  +  bx  +  c. 

(c)  Polynomials  of  four  or  more  terms  are  usually  factored  — 
By  grouping. 

As  the  difference  of  two  squares. 

(3)  Sometimes  an  expression  needs  to  be  rewritten  in  order 
to  show  the  type  of  factoring.  Before  concluding  that  an  ex- 
pression cannot  be  factored,  see  if  an  arrangement  of  terms  will 
bring  it  under  any  known  type  forms. 

(4)  Test  each  factor  to  see  if  it  can  be  factored  further. 

(5)  It  is  convenient  to  remember  that  x^  +  y^,  x~  -  xy  +  y-, 


x^  +  xy  +  ?/2,  a;2  +  xr/  - 

2/^ 

X2- 

-  xy 

-  ?/2  are  prime. 

MISCELLANEOUS 

Factor: 

EXERCISES 

1.  ax  -]-  ay  -{■  2abz. 

2.  2a2  _  2m\ 

3.  x'^  +  y'-2x7j. 

4.  a;2  +  llx  -  42. 

5.  x3  -  216y3. 

6.  343  +  1. 

7.  3x2  _  5a:  -  8. 

8.  27a'¥  -  ISab'^ 

122  FACTORING  [Chap.  Xlll. 

9.  (x  -  x2)  +  (x^  -  x^).  15.  3.r  -  5xif  -  Qxy  +  lOxyK 

10.  3x^  +  81.  16.  a'  +  2a'  +  1. 

11.  x^  +  19a;  +  18.  17.  X  -  8x\ 

12.  9  -  6a  +  a\  18.  16  +  Sab  +  a%\ 

13.  6^2  -  13x  -  28.  19.  6x2  ^  3^  _  3 

14.  a'  -  aK  20.  12  -  10a  -  2a2. 

21.  1  -  3a;  +  3x2  -  y.s^ 

Hint:  Collect  the  terms  thus,  (1  -  x^)  +  (Sx^  -  3x)  and  factor  expres- 
sions in  parentheses. 

22.  a^  +  2a2  +  4a  +  8.      25.  4a2  -  20a6  +  462. 

23.  X*  -  13x2  ^  36        26.  26x2  _  53^  _  5 

24.  x^  -  X  +  a;2i/  -  y.  27.   4x2  -  28x?/  +  Ady-. 

28.  a^fcV^  -  125c2.    Find  four  factors. 

29.  xHj  -  xy^. 

30.  24xy  +  x^  +  144x2;/''.    Y'md  four  factors. 

31.  ba^x^  -  5a^x2.  33.    11x2  +  9^-2. 

32.  16x  +  8a6x  +  a262a;.  34.   3a^  +  375a2. 

35.  8ai3  +  ai2. 

36.  6x2  _  _^(ct  +  2)  -  (a  +  2)2. 

37.  15x7  _  14^0  _  83.3_  44,   2xy«  -  lOx^?/^  -  28.tI 

38.  x2  -  a2  +  X  -  a.  45.   a2  +  (6  -  26x2)ay  -  2h'^xhf. 

5a'  -  5a~b  -  5ah  —  5a. 
1  -  a-b'^  -  x2?/2  +  2abxy. 
X"  -  (a  -  h)x  -  ah. 
22x2  +  59x  +  39. 
(2x  +  3)3  -  (2x  -  Z)\ 


(3x  +  yY  -  (2x  -  yf 
54.   4a362  +  36a26x  +  81ax2.       56.   x2  -  6x  -  247. 


39. 

25  -  (x2  +  2xy  +  1/2). 

46. 

40. 

36  -  x^  +  2xhf  -  y\ 

47. 

41. 

16a6  -  24a6x  +  9a6x2. 

48. 

42. 

343pY  -  729p6. 

49. 

43. 

51x2  _  453.  _  6 

50. 

51. 

50x2  ^  60x2/  +  18?/2. 

52. 

(2x  -f  3)  (x  +  y)  +  4(x 

+  ?y). 

53. 

4X3     _     ^^y     _     ^y.2y     ^    Qy2 

55 

Art.  82]  EXERCISES  123 

57.  7.r-  -  21c.r  -  280c-. 

58.  9a^  -  (ja^  +  a\ 

59.  (a-'  +  2ab  +  6'')  -  (.r^  -  2xy  +  if). 

60.  a-2  -  2.nj  +  if  -  49. 

61.  xij  -I  +  X  -  ij. 

62.  sr  -^  st  -  t'~  -  rt. 

63.  32.1-3  _  48^2^  ^  ig^j^2_ 

64.  X-  -  a-  -  b-  +  y-  -  2xy  -  2ah. 

65.  x^  +  x~  +  X. 

66.  n"-  +  4:mn  +  Am"^  -  16. 

67.  2.T?/  -  .T^  -  ?/  +  1. 

68.  2f  -  x{x  +  y). 

69.  8  +  (x  -  2)^ 

70.  .t"  -  nx^  -  90x2. 

71.  2a6a:'^  +  \Qa%x  -  28a^b. 

72.  8a-i-2  -  56a2.T?/  +  98a2|/2. 

73.  169.i-^  -  15Qxhf  +  SQxY- 

74.  27a^"  -  125.    Find  two  factors  only. 

75.  a'  -  a^b^-  -  a%^  +  b\ 

76.  x^  -21  -  9a-2  4-  27.r. 

77.  a;^  +  if  +  4z2  +  3.ry  -  if. 

78.  x^  +  ?/3  +  a:-  -  y'. 

79.  25.r^  +  10x3  _^  ^,2 

80.  28rt/>/i-  +  Mabmn  -  I2abm-. 

81.  8/y-.c-  -  SOifx  +  200y\ 

82.  3  -  x^-  +  3.r3  -  x"". 

83.  Cu--  4-  (  )  -  14.  Find  tlircMMliflVroiit  expressions  replac- 
ing tlie  liarcntheses  wliicli  will  permit  the  trinomial  to  1)C 
factored. 

84.  Write  down  at  random  three  trinomials  of  the  form 
ax^  +  bx  +  c  and  try  to  factor  them. 


124  FACTORING  [Chap.  XIII. 

Fill  the  following  parentheses  so  that  the  trinomials  will 
be  perfect  squares  : 


85.   49a V  +  14aa;  +  (  ). 

88.    (  )  +  ^a^xy  +  y"^ 

86.    (  )  +  \2pqrx  +  x\ 

89.   a;2  +  2xyz  +  (  ). 

87.    16x2  +  (  )  +  492/2. 

Factor  : 

90.    bx"  +  lOx  -  5x2  _  10. 

93.    (1  +  ay  -  (1  +  2a)2. 

91.   a;2  +  32/  -  3a;  -  xy. 

94.    {x  +  2)2  -  25(x  +  3)2 

92.    lUxYz^  +  ^4:xyz  +  1. 

95.   Write    down    three    perfect    square    trinomials    whose 
middle  term  is  4a6c. 

Factor  : 

96.  lOOax^  +  90a2x  -  90a\ 

97.  10x3  -  25X''  -  x2. 

98.  a2x2  +  b^x^  +  ay  -  by. 

99.  -  2cy  -  3bz  +  4:xy  +  cz  +  Qhy  -  2xz. 

100.  a2x2  -  62a;2  _^  ay  -  by. 

101.  x^  +  y^  +  x^y  +  xy^  +  X  +  y. 

102.  x^  +  7/3  +  x2  -  xy  +  y-. 

103.  a2  +  2a6  +  62  +  3a  +  36. 

104.  a*{a  +xy  -  (ax)\ 

105.  x^  -  xy  -  x-y  +  y^. 

106.  x2  -  2(a  -  6)x  -  4a6. 

107.  ax2  +  6x2  _|_  (jZy.  ^  53-j._ 

108.  Show  that  x2  -  16  is  a  special  case  under  ax2  +  6x  +  c. 

109.  What  values  have  a,  6,  and  c  when  x2  -  3x  is  consid- 
ered as  a  special  case  of  ax2  +  6x  +  c? 

Factor : 

110.  2xy  +  Zyz  +  6?/2  +  xz. 

111.  (x  +?/  +  2)2  -  1. 

112.  49^2^2  +  42pg  +  ?  =  a  perfect  square? 


Art.  82]  EXERCISES  125 

113.  ?/  -Sy  -  65.  117.  {a  +  by  -  (x  +  y)\ 

114.  x^  -  11.T2  _  12.  118.  8.r'  +  29x  +  15. 

115.  x^  +  6x3  ^  2.  119.  8x2  ^  (32x  +  15. 

116.  1728c3  +  64^3.  120.  Sx^  -  22x  +  15. 

121.  (a  +  6)2  -  (x  +  7/  +  zY. 

122.  (2  +  x)2  -  (2  +  X  +  yY. 

123.  4a6crf  -  24c(/x  -  Sabmn  +  ISmnx. 

124.  5.t3  +  1  _  a;2  -  5x. 

125.  2a.r2  +  daxy  -  2bxy  -  3by\ 

126.  9  -  (3  -  X  -  yY.  128.  Sx^  +  26x  +  15. 

127.  8x2  _^  34^  ^  X5         129.  x«  -  x"*  -  12. 

130.  (a  +  by  -  (a  -  by. 

131.  In  arithmetic  we  find  that  the  sum  of  the  odd  powers 
of  two  numbers  is  divisible  by  the  sum  of  the  numbers.  What 
case  of  factoring  comes  under  this  rule? 

132.  What  case  of  factoring  comes  under  the  rule  that  the 
difference  of  the  odd  powers  of  two  numbers  is  divisible  by  the 
difference  of  the  numbers? 

Factor  : 

133.  a  +  bx+  ay^  +  bxif. 

134.  lOx  +  lOi/  -  5ax  -  5ay  -  15a2x  -  \bahj. 

135.  x''"  -  1.    Find  two  factors  only. 

136.  3a2  -  6a6  -  4a  -  8&. 

137.  a^  +  1.    Find  two  factors  only. 

138.  20m2  -  9m?i2  -  20n2m2. 

139.  x2  +  2a2x  +  3x  +  6a2. 

140.  4932   _    (1    +  a;2  _^  2)2. 

141.  x2  +  ?/  +  0x2  _,.  aif  +  a2x2  +  ahf. 

142.  ax  -\-  ay  +  az  +  bx  -{- by  -\- bz  +  ex  +  cy  ^  cz. 


CHAPTER  XIV 
EQUATIONS   SOLVED   BY  FACTORING 


83.  Quadratic  equations.  If,  in  .r^  -  4.t  +  3,  we  substitute 
different  numbers  for  x,  we  find  different  values  for  the  expres- 
sion. Thus,  when  x  =  0,  the  value  of  a;-  —  4a:  +  3  is  3  ;  when 
a;  =  1,  its  value  is  0;  when  x  =  ^,  its  value  is  f .  We  give  a  table 
showing  the  value  of  the  expression  for  all  integral  values  of  x 
from  -4  to  +4. 

Suppose  we  ask  what  values  of  x  make  x^  -  4a:  +  3  equal  to 
zero.  By  referring  to  the  table 
we  see  that  the  expression  is 
zero  for  at  least  two  values  of 
X]  that  is,  when  a:  is  1  and  when 
a:  is  3.  We  have  really  asked 
a  question  equivalent  to  the 
following :  What  values  of  x 
satisfy  the  equation 

x2  -  4a:  +  3  =  0? 

This  equation  is  different  from 
the  equations  previously  con- 
sidered. In  the  left-hand  side  we  find  one  term  containing  x^, 
one  containing  x,  and  a  term  containing  no  unknown.  In 
other  words  the  equation  a:^  —  4a:  +  3  =  0  is  a  special  case  of  a 
class  of  equations  of  the  form  ox^  +  hx  +  c  =  0,  where  a  =  I, 
b  =  -4,  and  c  =  3. 

Equations  which  can  be  reduced  to  the  form  ax^  +  bx  +  c  =  0, 
in  which  a,  h,  and  c  are  any  numbers  whatever,  except  that  a 
cannot  equal  zero,  are  called  quadratic  equations. 
126 


X 

a;2  -  4x  +  3 

-4 

35 

-3 

24 

-2 

15 

-1 

8 

0 

3 

1 

0 

2 

-1 

3 

0 

4 

3 

Arts.  83,  84]  EXERCISES  127 

EXERCISES 

Make  tables  showing  the  values  of  the  following  expressions 
for  the  integral  values  of  x  from  -4  to  +4.  Indicate  if  possi- 
ble those  values  of  x  which  make  the  expressions  equal  to  zero. 


1.   X-  -  2>x  +  2. 

6. 

2x2  +  8^.  _  10. 

2.   x^  -x-2. 

7. 

2x-'  +  a;  -  1. 

3.   x^  -X  -  6. 

8. 

.t2  -  1. 

4.   x2  -  7a:  +  12. 

9. 

(x-  +  2)(2x-8) 

5.   x~  +  x  -  12. 

10. 

(.T-6)(x-3). 

Reduce  the  following  quadratic  equations  to  the  form 
ax-  +  6.r  +  c  =  0: 

11.  X-  -  3.r  =  2.  16.  2.r  -  5x'-  =  3x  -  7x^. 

12.  2.r-  -  4x  +  7  =  9.  17.  13  +x  -  4x-  =  2x'- +  x  +  1. 

13.  2  +x  =  x\  18.  2  -7x  +  x'-  =  2x-'  +  2. 

14.  3a;  =  7x^  -  4.  19.  a:^  -  32;  +  4  =  4  -  3a-. 

15.  7a;  +  3x-  -  1  =  x.  20.  px'^  +  qx  +  r  =  rx-  +  q-\-rx. 

84.  Factoring  applied  to  the  solution  of  quadratic  equations. 
To  solve  a  quadratic  equation,  it  is  not  necessary  to  proceed 
by  trials  as  in  Art.  83.  If  the  quadratic  expression  can  be 
easily  factored,  the  method  illustrated  by  the  following  example 
can  be  used. 

Example.     Solve  x^  -  4x  +  3  =  0.  (1) 

Solution  :  Factoring  the  left-hand  side  of  this  equation  we  find 

(x  -3)(x  -1)  =0.  (2) 

The  products  of  these  two  factors  is  zero  if 

X  -  3  =  0,  or  if  X  -  1  =0.  (3) 

From  (3),  x  =  3  or  1. 

The  equation  has  then  two  roots,  1  and  3.     Checking  by  substitution, 
(1)2  -4-1+3=0, 
and  (3)2  -4-3+3=0. 


128         EQUATIONS  SOLVED  BY  FACTORING    [Chap.  XIV. 

If  in  any  product,  any  factor  is  zero,  the  whole  product  is 
zero.  Conversely,  if  any  product  is  equal  to  zero,  some  factor 
of  that  product  must  be  zero,  and  any  factor  which  contains 
an  unknown  may  be  equal  to  zero.  Therefore,  in  solving  any 
quadratic  equation  in  which  one  member  is  zero  and  the  other 
member  can  be  factored,  we  find  values  of  x  which  make  each 
of  the  factors  zero.  That  is,  we  may  equate  each  factor  to  zero 
and  solve  for  the  unknown. 

Thus,  to  solve  2x='  =  7a;  -  5,  we  write  the  equation  in  the  form 
2x2  -  7a;  +  5  =  q.  Factoring,  we  have  (2a;  -  5)  (x  -  1)  =  0.  Equating 
each  factor  to  zero,  2x  -  5  =  0,  x  -  1  =  0,  and  we  find  x  =  |,  and  x  =  1. 

EXERCISES 

Find  two  solutions  for  each  of  the  following  quadratic 
equations  : 


1. 

(x  -  2){x  -  7)  =  0. 

10. 

2a;2  +  2a;  -  11  =  4a;  -  7 

2. 

(a;  +  3)(a:-  1)  =0. 

11. 

4a;  +  a;2  =  21a;  -  72. 

3. 

(2:r  +  5)(x  +  6)  =0. 

12. 

6  -  7a;  =  3a;2. 

4. 

(5x  +  6)(2a:-4)  =0. 

13. 

3a;2  -  14a;  =  a;^  +  60. 

5. 

(a:  +  l)(3x  +  7)  =0. 

14. 

x2  =  2a;. 

6. 

a;2  -  3a;  +  2  =  0. 

15. 

a;-^  -  4  =  0. 

7. 

a2  +  7a  -  30  =  0. 

16. 

4a;2  =  81. 

8. 

a;2  +  13a:  =  30. 

17. 

2a;2  +  5a;  -  3  =  0. 

9. 

a:2  +  lOx  +  24  =  -4a;. 

18. 

6  =  62^  +  352. 

Solve  first  for  x  and  then  for  m  in  Exercises  19  and  20. 

19.  2a;2  =  3ma;  -  m~. 

20.  a;2  -  9ma;  +  Um^  =  0. 

21.  a2  -  lah  +  106^  =  0.    Solve  for  a  and  for  h. 

22.  7a;2  +  13a;  =  8  +  3a;. 

23.  20.r2  +  11a;  -  3  =  0. 

24.  2a2  +  a  -  3  =  0. 


Art.  84]  PROBLEMS  129 

25.  X-  +  mx  +  Sx  +  3m  =  0.     Solve  for  x. 

26.  x~  -  mx  -  nx  =  0.    Solve  for  x. 

27.  X-  -  4a-  =  0.     Solve  for  x  and  then  for  a. 

28.  x-  +  ax  -\-hx  +  ab  =  0.    Solve  for  x. 

29.  x"  +  px  =  0.     Solve  for  x. 

30.  ax-  +  ax  +  bx  +  6  =  0.     Solve  for  x. 

31.  ax-  +  abx  +  x  +  b  =  0.      Solve  for  x. 

32.  aa;2  +  acx  +  fea;  +  6c  =  0.     Solve  for  x. 

PROBLEMS 

The  following  problems  involve  the  solution  of  quadratic 
equations;    find  all   solutions   possible  for 
the  equations  and   determine   whether   or 
not  each  solution  is  a  reasonable  result: 

1.  In  a  right-angled  triangle  the  longer 
leg  is  two  feet  shorter  than  the  hypotenuse 
but  two  feet  longer  than  the  shorter  leg. 
What  is  the  length  of  the  longer  leg? 

Solution  :  Let                       x  =  the  length  of  the  longer  leg. 

Then  x  -  2  =  length  of  the  shorter  leg, 

and  X  +  2  =  length  of  the  hypotenuse. 

From  page  80  (x  +  2)=  =  x'-  +  {x  -  2)2. 

(See  Fig.  20,  also  Fig.  15). 

Hence  x'^  +  ix  +  A  ^  x'^  +  x-  -  4.x  +  4, 

or  x2  -  8x  =  0. 

Whence  x  =  0,  or  x  =  8. 

The  answer  x  =  0  must  be  cast  aside,  for  it  has  no  interpretation  when 
appUed  to  the  problem  in  question. 

2.  In  a  right-angled  triangle,  the  short  leg  is  two  feet 
shorter  than  the  hypotenuse  and  one  foot  shorter  than  the 
longer  leg.     Find  the  length  of  the  short  leg. 

3.  A  positive  number  when  multiplied  by  a  number  5 
times  as  large  becomes  405.     What  is  the  number? 


130  EQUATIONS  SOLVED   BY  FACTORING     [Chap.  XIV. 

4.  The  square  of  a  certain  number  plus  twice  the  number 
itself  is  equal  to  eight  times  the  number.     Find  the  number. 

5.  The  product  of  two  consecutive  integers  is  306.  What 
are  the  numbers? 

6.  The  sum  of  the  squares  of  two  consecutive  integers 
is  41.     What  are  the  numbers? 

7.  The  sum  of  the  squares  of  three  consecutive  integers 
is  50.     What  is  the  smallest  of  the  numbers? 

8.  The  square  of  a  number  is  20  more  than  the  number 
itself.     What  is  the  number?     Is  there  more  than  one  answer? 

9.  The  area  of  a  rectangle  is  18  square  feet.  The  length 
is  one  foot  longer  than  4  times  the  width.  What  are  the  di- 
mensions of  the  rectangle? 

10.  A  rectangular  floor  is  4  feet  longer  than  it  is  wide,  and 
its  area  is  320  square  feet.     What  are  its  dimensions? 

11.  The  perimeter  of  a  rectangular  field  is  60  rods,  and  its 
area  is  200  square  rods.     What  are  its  length  and  width? 

12.  Make  up  a  rectangle  problem  whose  solution  will  in- 
volve a  quadratic  equation. 

13.  A  photograph  is  one  inch  longer  than  it  is  wide.  It 
is  mounted  on  a  card  so  that  there  is  a  1-inch  margin  on  all 
sides.  The  total  area  of  the  margin  is  2  square  inches  greater 
than  the  area  of  the  photograph.  What  are  the  dimensions 
of  the  photograph? 

14.  It  takes  96  square  inches  of  paper  to  cover  a  cube. 
What  is  the  length  of  one  edge  of  the  cube? 

15.  The  dimensions  of  a  closed  rectangular  box  are  con- 
secutive integers.  The  entire  outside  surface  of  the  box  is 
52  square  inches.     What  are  the  dimensions  of  the  box? 

16.  A  paving  brick  is  4  inches  longer  than  it  is  wide.  The 
thickness  is  4  inches.  The  volume  of  the  brick  is  128  cul)ic 
inches.     What  are  the  length  and  width  of  the  brick? 

17.  A  rectangular  box  is  5  times  as  wide  as  it  is  deep  and 
twice  as  long  as  it  is  wide.  The  total  surface  of  the  box  is  130 
square  inches.     What  are  the  diinensions? 


Art.  S4]  PROBLEMS  131 

18.  A  rectangular  solid  is  twice  as  long  as  it  is  wide,  and 
the  width  is  3  inches  more  than  the  depth.  The  total  surface 
is  160  square  inches.     Find  the  dimensions  of  the  solid. 

19.  Make  up  a  problem  about  a  rectangular  solid  or  brick 
whose  solution  will  involve  a  quadratic  equation. 

20.  A  club  had  a  dinner  that  cost  $60.  If  there  had  been 
4  persons  more,  the  share  of  each  would  have  been  50  cents 
less.     How  many  persons  were  there  in  the  club? 


CHAPTER  XV 

HIGHEST  COMMON  FACTOR  AND  LOWEST  COMMON 
MULTIPLE 

85.  Greatest  common  divisor  in  arithmetic.  The  largest 
integer  contained  as  a  factor  in  two  or  more  integers  is  called 
in  arithmetic  the  greatest  common  divisor.  This  number  is 
easily  found  by  separating  the  numbers  into  their  prime  factors 
and  multiplying  together  those  found  in  each  number.     Thus,  in 

12  =  2  •  2  •  3,     18  =  2  •  3  •  3,     30  =  2  •  3  •  5, 

the  greatest  common  divisor  is  2  ■  3  =  6. 

86.  Highest  common  factor.  In  algebra,  a  number  or 
expression  which  is  a  factor  of  each  of  two  or  more  expressions, 
is  called  a  common  factor.  Thus,  in  lOx'^y,  4xy,  Sxi/,  the  com- 
mon factors  are  ±2,  ^x,  ^y,  =^xy,  ^2xy.  Usually  the 
positive  factors  alone  are  considered.  The  factors  2,  x  and  y 
are  the  common  prime  factors. 

The  product  of  all  the  common  prime  factors  of  two  or 
more  expressions  is  called  their  highest  common  factor. 
(H.C.F.).  For  the  above  expressions  lOxhj,  4:xy,  8xy^,  the 
H.C.F.  is  2xy. 

Two  expressions  that  have  no  common  factor  except  1,  are 
said  to  be  prime  to  each  other. 

In  algebra,  the  word  "highest"  is  preferred  to  the  word 
"greatest"  in  connection  with  common  factors.  The  highest 
common  factor  is  the  common  factor  which  contains  the  highest 
number  of  prime  factors.  In  the  above  example,  2xy  is  the 
highest  common  factor,  but  if  .r  =  1  and  y  =  ^  it  is  not  the 
greatest  common  factor. 

132 


Art.  8G]  EXERCISES  133 

EXERCISES 

Find  the  highest  common  factor  of  the  following  sets  of 
expressions : 

1.  8«%V,  4a%V,  lQa'-bh\ 

The  common  prime  factors  are  2,  2,  a,  a,  b,  h,  c,  c.     Hence 
the  H.C.F.  is  4a'-lfc'. 

2.  lOx^yh^,  5xy^z^.  5.    bmn-jp^q^,  XQm^n^,  20mnpq. 

3.  7a63c^  21a*62c3.  6.    12a'b^,  Qa'b',  18a»6^  9a'b\ 

4.  Sa%  6ab,  Zb".  7.   8xy,  IQxY,  ^x^U,  "if- 

8.  Uabc,  8a%^,  la?b~c\  2a^b-K 

9.  Sx^y,  QxY,  12xV,  24xh/- 

10.  2a^  -  2ab^,  4a^  +  8a-b  +  iab^. 

Solution:    2a'  -  2ab^  =  2a (a^  -  6^)  =  2a (o  +b){a  -b). 
4a3  +  8a%  +  4ab^  =  2  •  2  •  a(a  +  6)  (a  +  6). 
The  common  prime  factors  are  2,  a,  (a  +b). 
Hence,  the  H.  C.  F.  =  2a(a  +  b)  or  2a2  +  2a6. 

11.  x^  -  y-,  x~  -  2.ry  +  if. 

12.  a:^  -  y^,  x-  -  y-. 

13.  x^  +  if,  xHj  +  xy^. 

14.  x^  -  if,  x^  +  x-y  +  xy-. 

.  15.  a^  +  6\  2a2  -f-  4a6  +  26^,  a^  -  b"^. 

16.  ax  -  ?/  +  a;i/  —  a,  ax-  +  x-y  -  a  —  y. 

17.  x-(/  -  ?/  -  x-2;  +  2,  X(/  -  X2  -  ?/  -  z. 

18.  1584,  1728. 

19.  861,  615,  984. 

20.  156,  130,  182. 

21.  9  -  X-,  X-  -  X  -  6. 

22.  X-  -  2x,  x2  -  X  -  2,  2x2  -  X  -  g 

23.  x-*  -  x'-y-,  Sx"*  -  x'-i/  -  2xy-,  x^  -  2xhj  +  xy"-. 

24.  a^x  +  (C-y  +  a^x  +  a6i/,  2ax-  -  2ay-,  Sax^  +6ax!/  +  Zay^. 

25.  a6x2  -  abxy  +  a6(/-,    abx-  +  aby"^,    abx-  +  abxy  +  b'^y^. 


134  H.  C.  F.  AND  L.  C.  M.  [Chap.  XV. 

87.  Lowest  common  multiple.  An  expression  which  con- 
tains each  of  two  or  more  given  expressions-  as  a  factor  is  called 
a  common  multiple  of  the  expressions.  Thus,  if  x,  xy,  10a:, 
2y,  5xy~  are  given,  then  lOxy^  is  one  common  multiple.  There 
are  many  more,  for  example,  20x~y^,  lOOxy"^,  AOx^y^z^. 

Among  all  the  common  multiples  of  a  given  set  of  expres- 
sions, there  is  one  which  is  most  important.  It  is  called  the 
lowest  common  multiple  (L.C.M.).  The  lowest  common  mul- 
tiple of  two  or  more  expressions  is  the  product  of  all  their 
different  prime  factors,  each  factor  being  used  the  greatest 
number  of  times  it  occurs  in  any  of  the  expressions.  For 
example,  consider  the  expressions  x,  xy,  10a;,  2y,  5xy^.  In 
these  the  different  prime  factors  are  x,  y,  2,  5.  The  factors  x, 
2,  and  5  occur  only  once  in  any  expression.  The  factor  y  occurs 
twice  in  5xy^,  since  we  may  write  it  5xyy.  The  L.C.M.  is  then 
2  •  5  •  X  •  y  •  y  =  lOxy-. 

EXERCISES 

1.  What  is  the  least  common  multiple  of  two  or  more 
numbers  in  arithmetic? 

2.  In  connection  with  common  multiples  in  algebra  why 
should  the  word  "lowest"  be  preferred  to  the  word  "least"? 

Find  the  L.C.M.  of  the  following  sets  of  expressions  : 

3.  xy,  Saxhf,  2a?.  8.   x'^y^,  a;"-V^  x"~hj\  x"'^^y. 

4.  3xyz,  5ax,  lOah^.  9.    (x  +  y),  x^  +  y-,  x^  -  y'-. 

5.  \2a%c^,  ^alrx",  '^ah(?x.     10.   x^  -  if,  x^  +  2xy  +  y\ 

6.  ln^m\  V^anhn,  2amn.     11.   x^  +  3a;  +  2,  rc^  +  4a:  +  3. 

7.  a;",  xy,  xy"\  12.   Qx^  +  13a;  +  6,  ix-  -  9. 

13.  3x2  _  lox  +  8,  a;2  -  4a;  +  4. 

14.  ah  -  b^,  a2  +  ab,  ah  +  b\ 

15.  6a;2  -  5x  +  1,  Sa;^  -  6a;  +  1,  lO-r^  -  7a;  +  1. 

16.  3a;2  +  5x  +  2,  3.r2  +  8a;  +  4,  3a;2  -  4a;  +  4. 

17.  x3  -  y\  x"  -  1/2. 

18.  a;^  +  y^,  x^  ^  xy. 


Akt.  S7]  exercises  135 

19.  x^  +  xif,  x^  -  x'-y  +  xif,  X  +  ij. 

20.  a  -  h,  (a2  -  b-)~,  («  +  ^)-''- 

21.  ax-  +  ax,  ax'  +  Sax  +  2a,  x~  +  4x  +  4. 

22.  S{ab~  -  ¥)-,  2b(a-b  -  x'Y,  6{a%^  -  a'). 

23.  X-  -  xy  -  5x  +  5y,  x-  -  lOx  +  25,  x^  -  25. 

24.  a.T  +  a  +  2x  +  2,  ha^  +  7a  -  6,  5ax  +  5a  -  3x  -  3. 

25.  6a.T  +  9a  +  4x  +  6,  2ax  -  4x  +  3a  -  6,  30^  -  4a  -  4. 

26.  a%  -  ab\  7a'-  +  lab  +  7b'-,  Ua^b  +  Ua^b^  +  Uab\ 

27.  1  -  m  +  n  -  mn,  1  -  m  -  x  +  mx,  \  +  n  -  x  -  nx. 

28.  x*  +  X-  -^  X  +  \,  x^  +  X,  a;2  +  2x  +  1. 

29.  ax  +  y  -by  +  ay  -  bx  -\- x,        ax  +  ay  -  bx  -  by, 
a-  —  ab  +  a  -  ah  +  b-  -  b. 

30.  ax  +  a  -  bx  +  ay  -  b  -  by,  {x  +  ij)-  +  x  +  y, 
ax  +  ay  -  by  -  bx. 

31.  a^  +  ¥,  ac  -  be  +  ad  -  bd,  a^c  +  abc  +  b-c  +  a-d  +  abd 
+  bH. 


CHAPTER  XVI 
FRACTIONS 

88.  Fractions  in  arithmetic.  If  a  unit  be  divided  into  7 
equal  parts  and  3  of  these  are  taken,  we  denote  tiie  amount 
taken  by  f .  This  fraction  may  also  be  understood  to  mean 
3^7. 

The  fraction  f  is  thus  the  answer  to  the  questions  : 

1.  What  part  of  7  is  3? 

2.  What  is  3  ^  7? 

3 
But  a  fraction  such  as  ^  cannot  be  regarded  in  the  first  of  the 

two  ways  mentioned,  since  a  unit  cannot  be  divided  into  2\ 

3 
equal  parts.     The  fraction  ^  is  thus  taken  to  mean  3  -h  2 j ; 

3 
that  is,  the  fraction  ^  is  an  indicated  division. 

-^4 

89.  Fractions  in  algebra.  Similarly,  in  algebra  a  fraction 
is  an  indicated  division  in  which  the  dividend  and  divisor  are 
algebraic  expressions.     Thus, 

6  a    -3         ,  a2  +  6^ 

7  0    4a  x^  +  y^ 
are  fractions. 

The  dividend  is  called  the  numerator,  and  the  divisor  the 
denominator  of  the  fraction.     The  two  are  often  called  the 

terms  of  the  fraction.     The  fraction  -  is  read  "  x  over  y,"  or 

y 

"x  divided  by  y." 

The  rules  for  fractions  used  in  arithmetic  apply  in  algebra. 
In  particular,  much  use  is  made  of  the 
136 


Art.  S!)]  EXERCISES  i;J7 

Principle.  The  numerator  and  the  denominator  of  a  fraction 
may  be  midtiplied  or  divided  by  the  same  number  without  changing 
the  value  of  the  fraction. 

Thus, 
Similarly, 


and 


2      2-2      4 

3      3-2      6* 

14      14  -^  7      2 
21      21  ^  7  ~  3' 

a2 

-b-^ 

(a2  -  62)  -=-  (a  -  6) 

Li* 

(a  -  6)2     (a   -  6)2  H-  (a  -  6)      a  -  6" 


EXERCISES 


1.  State  two  meanings  of  -  ;    of  ^  ;   of  r  ;   and  of 

if  fl,  b,  and  c  denote  integers. 

2.  In  his  four-year  course,  a  student  spends  for  Ijooks  a 
dollars  the  first  year,  b  dollars  the  second,  c  dollars  the  third 
and  d  dollars  the  fourth.  What  fraction  denotes  the  average 
expense  per  year? 

3.  What  part  of  a  piece  of  work  can  a  man  do  in  one  day, 
if  he  can  do  the  entire  work  in  6  days?  In  5|  days?  In 
a  days? 

4.  The  grades  of  a  student  in  two  subjects  differ  by  5 
points.     The  lowest  grade  is  x.     What  is  the  average  grade? 

By  multiplying  or  dividing  both  numerator  and  denom- 
inator by  certain  numbers,  replace  the  following  fractions  by 
equal  fractions.     Give  at  least  three  solutions  for  each  exercise. 


5. 

7 
8' 

8. 

f 
a' 
T 

11. 

x  +  y 
x-y 

14. 

-14 
Gx 

6. 

4 

9" 

9. 

X 

12. 

a 
a-b 

15. 

a% 
a^b' 

7. 

13 

2' 

10. 

a- 
62  ■ 

13. 

aH 
ab^' 

16. 

cd 
cdl^' 

138  FRACTIONS  [Chap.  XVI. 

90.  Division  by  zero.  The  numerator  of  a  fraction  may 
be  any  number  whatever,  including  zero.  In  the  case  of  the 
denominator,  however,  we  have  one  exception.  The  denom- 
inator camiot  be  zero,  for  division  by  zero  is  excluded.  That  is, 
an  expression  with  denominator  zero  is  not  considered  as  a 
number.     Care  must  be  taken  not  to  give  such  values  to  the 

X  +  4 

letters   as  will  make  the  denominator  zero.     Thus, is 

X  -  3 

a  number  for  any  value  of  x  except  a;  =  3.     For  a;  =  3,  we  have 

7 


,  which  has  no  meaning. 


EXERCISES 


Give  the  values  of  x  for  which  the  following  expressions 
have  no  meaning : 

x  +  2  3a; +  4 

'  x-2'  3a;  -  4" 

X  .     1 


a;  +  1  X 

a;  +  1  ^    X  +  4 


1 

a:(a;  -  1) 

3  -5a: 

3a;  +  5a;2 

2 

(2a;-  l)(a:  +  2) 


91.  Signs  in  fractions.  In  any  fraction,  there  are  three 
signs  to  consider  :  the  sign  before  the  fraction,  the  sign  of  the 
numerator,  and  the  sign  of  the  denominator.     Thus, 

Historical  note  on  fractions.  Fractions  offered  very  great  difficulty 
to  the  ancient  nations.  In  their  operations  with  fractions,  the  Babylo- 
nians reduced  all  fractions  to  the  denominator  60.  Similarly,  the  Romans 
reduced  fractions  to  the  denominator  12.  The  Egyptians  and  the  Greeks 
reduced  fractions  to  the  same  numerator.  In  fact,  Ahraes  (see  p.  80) 
seems  to  have  restricted  the  term  fraction  to  those  with  numerator  equal 
to  1.  Other  fractions  such  as  |  were  expressed  as  the  sum  of  fractions 
with  numerators  equal  to  1.  Thus,  "  \  and  \  "  took  the  place  of  f .  This 
probably  seems  rather  awkward  to  us,  but  it  shows  that  fractions  offered 
real  difficulties  to  the  early  students  of  mathematics. 


Art.  91]  SIGNS  IN  FRACTIONS  139 

+  +7'        +9'        -9' 

have  the  three  signs  attached. 

Since  the  names  numerator,  denominator,  and  fraction  mean 
dividend,  (Uvisor,  and  quotient  respectively,  the  laws  of  signs 
for  division  must  hold  for  fractions.     Hence,  we  have 

~V  ~  ~h'  -b~  ~b'  1^6"+ 6" 
Further, 

In  words,  we  have  : 

(1)  A7iy  two  of  the  three  signs  affecting  a  fraction  may  he 
changed  without  changing  the  value  of  the  fraction. 

(2)  The  sign  of  a  fraction  is  changed  by  changing  the  sign  of 
either  numerator  or  denominator  alone. 

+3  —3  —3  +3 

Thus,  —ij  and  — ^  are  both  positive  ;    — ^  and  —  are  both  negative. 

In  changing  the  sign  of  the  numerator  (or  denominator) 
when  it  is  a  polynomial,  be  careful  to  change  the  sign  of  each 
term  of  the  polynomial  inckuling  the  first. 

Thus,  changing  the  sign  of  the  fraction 
X  -y 
Sx  +y 

by  changing  the  sign  of  the  numerator  gives 


3x  +y 


EXERCISES 
Reduce  to  equal  fractions  with  j^ositive  signs  in  Ijolli  iiumera- 
tor  and  denominator  : 

1.   ^.  2.    =^-  3.   ^^!^- 

*•    -6  9  -c 


FRACTIONS 

[Chap.  XVI 

4. 

-4c 
-a-b 

e           6 

^-         -x  -  2" 

^'         -x-5 

5. 

-x-2 
5 

-771-2 

n  +  5 

9          -«-^ 

10. 

Show  that  - 
a 

-a           a         ^. 

— 5  =  ^ Give 

-6      b  -  a 

reasons  for  each  step. 

11. 

Show  that  3- 

5                  5 

-  y          ?/  -  1 

12. 

Showthat  ^ 

2a                    2a 

6(6  -  a)  b{a  -  6) 

92.  Reduction  to  lowest  terms.  A  fraction  is  in  its  lowest 
terms  when  no  factor  except  1  is  contained  in  both  numerator 
and  denominator. 

4x  —  1                                                            ^           x"^  —  X 
Thus,  ■=  and  r-  are  in  their  lowest  terms,  while  -x  and  —. are 

not.  To  reduce  a  fraction  to  its  lowest  terms  we  use  the  principle  of  Art. 
89  and  divide  both  numerator  and  denominator  by  the  factors  common 
to  both.     For  example, 

x^  -  X  _  x{x  -  1)  _  X  -  1 

x^  +  X  ~  x{:x  +  \)  ~  a;  +  1 

EXERCISES 

Reduce  the  following  fractions  to  lowest  terms  : 

7. 
8. 


1. 

21 

28' 

2. 

4a62c3 

2abc 

3. 

250 
375" 

2x3  _ 

2^2 

2x'  +  2x^ 

3x'  -3 

x  +  1 

3x 

3x^  + 

Qx'  +  9x 

10. 

ax^  +  axy 

2{x 

-«) 

21  +  lOx  +  x^ 

.t2-9 

(.r- 

-  3)  (.r  +  4) 

16 


ax^  -  axy^ 


Art.  93]  CANCELLATION  141 

93.  Cancellation.  The  process  of  dividing  the  numerator 
and  denominator  by  the  factors  common  to  both  is  called 
cancellation  by  division.  It  is  merely  an  application  of  the 
principle  of  Art.  89.  The  procedure  is  illustrated  by  the 
following  examples. 

Example  1.     Reduce    .^  .„,  ^  to  lowest  terms. 

Solution  :  ^^^^  =  __^_  =  __. 

Example  2.    x^  -  xy^     ;x{^&-=^(x  +y) 


X'  -  xV  .X(»-=T)(a;2  +  xy  +y')       x{x^  +  xy  +  y^) 

X 

Cancellation  can  be  used  only  with  factors  of  the  numerator 
and  denominator. 

Thus,  in  the  fraction  -    -, —^  we  may  cancel  the  2's,  but  in  " 

2  •  (X  +  ?/)  -^  2  +y 

we  cannot  cancel  the  2's  ;   for,  here  2  is  not  a  factor  of  the  numerator 

7      2+5 
or  denominator.     To  illustrate,  ^  =  ^ ^-     To  strike  out  the  2's  gives 

^-     But  -  is  not  equal  to  ^■ 

D  b  D 

EXERCISES 

Reduce  to  lowest  terms: 

1.  ^.  6. 

2.  — ^  •  7. 

7,5a¥(^d* 
2bhc'd?  ' 

28xyh'  g 

^-   2x  +  2xy  7  -  7x 


4:+4X 

6  +  3x 

9+Sx 

2 

2  +  x 

4  +  2x 

2  +  x 

14 -7a; 

142  FRACTIONS  [Chap.  XVI. 

11.    ^ ;;"•  lo. 


Sx  -  Qy  ax  +  3x  +2a  -\-Q 

12. 2+M^.  19.  ^'  -  y- 

Uy 
13.    *^^+/^^Jl.  20. 


14.     „   '         ^     2-  21. 

x^  -  2xy  +  y^ 


15. 


i"^  +  8a  +  2 
16a2  -  4    *  x"^  +  6x  +  9 


a;'^ 

-If 

a^ 

-a¥ 

a^ 

-a^¥ 

4a2x2  +  8a2x  +  Sa^ 

2ax  +  3a 

ax 

+  3a  -bx  - 

-36 

mn?  -  m  36aa;  +  45a?/  -  4rc  -  5y 

?w2n2  _  2m2w  +  m2  ■  "  Sx  +  lOy 

a^b  -  36b  12ax  -  6aj/  -  50cx  +  25c?/ 

a2  +  6a  '  2x^  +  xy  -y^ 

94.  Reduction  to  common  denominator.  Two  fractions 
are  said  to  be  equivalent  when  one  is  the  result  of  multiplying 
or  dividing  both  terms  of  the  other  by  the  same  number. 

2  6     cc^  x^    ctx  (I 

Thus,  ^  and  ,— ,  —  and  — ,  —z  and  ->  are  pairs  of  equivalent  fractions. 
'  5  15    yz         xyz    x^         x 

Any  set  of  fractions  may  be  changed  into  a  set  of  equivalent 
fractions  all  having  a  common  denominator. 

Thus,  ~,  -X,  — r —  may  be  written  ^rr-,   ^,    -^-51 -■ 

'  y    Q        b  Qby    6by  Qby 

This  may  be  done  in  many  different  ways,  but  when  the 
common  denominator  is  the  lowest  common  multiple  of  all  the 
denominators  the  process  is  called  reduction  to  the  lowest  com- 
mon denominator  (L.  C.  D.).  The  lowest  common  denominator 
of  any  set  of  fractions  is  the  L.  C.  M.  of  the  denominators. 


Arts.  94,  95]  EXI':RCIS1':S  143 

EXERCISES 

Reduce  the  following  sets  of  fractions  to  the  lowest  common 
denominator : 

1    1,3,7.  o    ^ L_ 

'258  '  y-l'  y{l  -y)' 


o     3     3     5 

2-    TZ'  -'  -x:-  9.   X,  y 


X  y 


4a    a    3a  x  -  \    I  -  x 

a  +  h    a  -b  m  9    "    «  +  1    «  +  2 

dab'  '     a'b  '  ■""•  ^'6'      6^    '      6^    * 

^a  +  6a-6  ..a:  +  l         x  -  I 

4.    f)  — — r-  11.  -,  2-,  -• 

a  -  b    a  +  b  x  -  1         x  +  I 

5       2x               3  ^2  1    ^,  ^,  -1    ^. 

■   a;  +  2'  x-  +  5a:  +  G"  '     '6'  a    ab'   2  ' 

x        xy  y                    ^^  X        1             y 


X-  -  y"^     \     X  -  y  y    xy  -  y    xy  -  X 

_a  +  l        a        a  -  \  ^^        x  y  1 

7.     7>    7' 7-  14. 


15. 


x-\x-\-\x-l  X  -  y    y  -  X    -  X  -  y 

2  3 

ce  +  5a  +6'  2 (a-'  +3a  +  2)' 
a;  y  I  \ 

16.  -; x>     —:: ::)    »     • 

x^  -\-  y^    x^  -  y^    X  —  y    x  +  y 

17.  a  +-,  VI  +  4|- 

c 

2 

18.  5^  +  a,  a  +  6  -  h  ;rT^' 

a  +  o 

95.  Addition  and  subtraction  of  fractions.  In  adding  or 
subtracting  algebraic  fractions  we  proceed  as  in  arithmetic. 
Just  as  3  feet  +  4  feet.  =  7  feet, 

3^4  7 

'°  11 +  11  =n' 

3      4  7 

and  -  +  -       =  — 

a      a  a 


144  FRACTIONS  [Chap.  XVI. 

EXERCISES 
Perform  the  following  additions  and  subtractions: 

1. 


1        2 

7+7- 

6. 

X-  _  6      26_ 
a      a       a 

9^  +  A+  A 
12        12  ^  12" 

7. 

6             4            2a: 

x+l       X  +  1  ^  a;+  1 

1      2      3 

a  +  a+a 

8. 

a  +  5      a~h 
2             2 

a      2a      1 

8  ~  8  ^  8* 

9. 

1  +  I  _  8. 

,.2     +    ^2            ^2 

tn       m       m 

10. 

X  +  i/  ^  2.r      ?/ 
n          n       n 

To  add  expressions  involving  different  units,  as  3  feet  and 
24  inches,  we  reduce  them  to  the  same  unit.     Thus, 
3  feet  +  24  inches  =  3  feet  +  2  feet  =  5  feet. 

Similarly,  to  add  f  and  §  they  must  be  reduced  to  the  same 
denominator.     Thus, 

9        8       17 
4  +  3  = 
In  like  manner. 


4  "^  3      12  +  12      12 


a      c  _  ad      i>c  _  ad  +  he 
h^~d~hd^hd~      bd 

Similarly,  for  subtraction.     Thus, 

_y_   _  ^  =        47/        _  Sxij  -  3x    ^  4:y^  -  (3xy  -  Sx) 
y  -  1        4y~  4y(y  -  1)       ^y{y  -  1)  ^y{y  -  1) 

_  4/  -  Sxy  +  Sx 

My  -  1) 

When  this  is  put  in  the  form  of  a  rule,  we  have  : 
To  add  fractions,  reduce  the  given  fractions  to  a  common  de- 
nominator, add  together  the  new  numerators  and  place  this  sum 
over  the  common  denominator. 


Arts.  95,  96]  EXERCISES  145 

A  similar  rule  holds  for  subtraction  except  that  we  subtract 
one  numerator  from  the  other  instead  of  adding.     . 

EXERCISES 

Perform  the  following  atlditions  and  subtractions  : 


1. 

hhh 

11. 

1               1 

.T  -   1         X  +  \ 

2. 

2a      da 

5  +  4  ■ 

12. 

1           1               a 

a       1  +  a       a2  -  1 

3. 

3.r       X       1 

5  +  10  +  3  ■ 

13. 

a-b      26  -  5c      1 

3c     +      7c       +2- 

4. 

2/  -  1    ,  ?/  -  2 

4       '       5     ■ 

14. 

X  -  ij      z  -  X      y  -  z' 

5. 

2       5        7 

a  +  2a  +  3a  ■ 

15. 

1111, 

X^         7^         X^         X 

6. 

5a;  +  6      3a:  +  a 
14             21 

16. 

a  +  1            2a 
(a  -  1)2  '    a2  -  1 

7. 

5       3      1 

^^  +  ^  +  2- 

17. 

1                1            a- -5 

3a:  -  3  '  2x  +  2  '  6x  -  6 

8. 

y      a      X 

18. 

X              xy               xy 

x  +  y      ix  +  yY      {x  +  yY 

9. 

7        5       6 

a^      ab      b^ 

19. 

2  -  2a         a  +  1            1 

(a  _  1)3   1    (a  _  1)2      a  -\ 

10. 

l-l-l- 

20. 

.7fi-^-r--T- 

96.  Multiplication  of  fractions.  As  in  arithmetic  the  pro- 
duct of  any  number  of  fractions  is  the  fraction  ichose  numerator 
is  the  product  of  the  numerators  and  whose  denominator  is  the 
product  of  the  denominators. 


146  FRACTIONS  [Chap.  XVI. 

^,        ..,,,.  1    2    5     10      5 

Thus,  in  arithmetic  2  '  3  '  6  =  36  =  Is' 

In  algebra, 

2  ,  X  +3  2x  +  2  2(^f^f^  •  2(-»-rT)  4 


X     x+1      x2+4x+3       x(«-rl)(«-r^(x  +  1)      x(x  +  1) 

Expressions  not  in  fractional  form  can  always  be  considered 
as  fractions  with  I  for  denominators. 


EXERCISES 

Perform  the  following  multiplications  and  reduce  the  an- 
swers to  the  lowest  terms: 

.     J^    ^    3  X        X  +  y    X  -  y 

•   32    21    2  x  +  y'      X     '  x  +  y 

2     z8    10    ^  10    ^    ^    ay 

25     3     16  •      "•   by'  a^x^'-hx 

4      -9    30     6  x-\-y    ^^       ^^  ^' 

^     Sab    4:ax  _^    (2x 

2xy  96 
562  8^ 
2a2'  25 


5.  «L:.|L^  13.  r^^Y-M-'.a. 


6.  ^.  ^.  ^•^.  14.  ^./^V.  <^^  +  ^) 


bed 


'   y    z    X  '       X     '      y     '  y' 

_        a        2a -2  ^^    ^  +  U         1         /          n 

8.    r  •  ^ 16.    —^^^  • -,  •  (x  ~  y). 

a  -  1        3a  X  -  y    x^  -  y^    ^        ^' 

x^  +  2xy  +  y^    a-  +  2a6  +  6^ 

a^  -  6^  x"^  -  7/ 

^g     (g  +  1)  (g  +  2)     (g  +  2)  (g  +  3)        1 

g  +  3  rt  +  1          *  g  +  2" 


Art.  96]  EXERCISES  147 

ax  +  x^  x~  -  a^ 


19 


21. 


ax  -  X-    X-  +  Sax  +  2a^ 
x^  +  X    xy  -  X    y  +  \ 
y--y    xy  +  X    x  +  I 
rt-  -  1         a^  +  4a  +  a2 


a~  +  5a  +  Q    a^  +  2a  +  1 


22.  '-±l.-l±y^,.(r--2nj +  ,/-). 
X  -  y    x^  -  y^ 

^^'   a-^h\    h      '^a  +  bj 
24.    [m )  Im  +  - 


'•("-S(' 


tnj    mr  —  4: 


MISCELLANEOUS  EXERCISES 

In  the  following  Exercises,  1 — 6,  the  letters  are  assumed  to 
be  positive  integers. 

11  .  13 

1.  Which  is  larger,  -  or  —  ?     Why?     -  or  -?     Why? 

X         oX  X         X 

2.  Which  is  larger,  -  or  7;—?     Why?     -  or  — ?     Why? 

^      X      Sx  -^       X        X  *^ 

3.  AVhich  is  larger,  j 7  or r?     Explain. 

4.  Which  is  larger,  -  or  —^ — ?     Explain. 

o  o 

3  3. 

5.  Which  is  larger,  -  or 7?     Explain. 

''a       a  +  1  ^ 

6.  Which  is  larger,  -  or  -? 

X      y 

7.  Does  -  =  —  for  all  values  of  x  and  a? 

X     Sx 

8.  Does  -  = ?,  for  all  values  of  x  and  a ;  for  anv  values 

x      x  +  3 

of  X  and  a? 


148  FRACTIONS  [Chap.  XVI. 


9.   Is  it  true  that 


X  - y      X  +  y 

10.  Does  77^  =  7-J-?     Explain. 

IQy      Uy 

11.  Does r  = -^ r^-^     Explain. 

-a  -  b        a^  -  hr 

12.  Which  will  give  the  greater  product,  if  x  and  y  are  posi- 
tive integers,  to  multiply  a  positive  number  by  ^  or  to  multiply 

X 

the  same  number  by  ^r-?     Explain. 
3?/ 

13.  Arrange  in  order  of  size,  if  a  and  x  are  positive,  (a)  if  x 
is  greater  than  1;  (6)  if  x  is  less  than  1. 

a        a  a  a 

X    X  -\-  l'   X  -\-2J  X  -  I 

14.  Fill  the  blanks  in  the  following  : 
a      ka       a  -  a  a       ? 


b       ?    a-b        ?     6      ab' 
X  ?  1  ?  X  2x  +XZ 


X  -  y      X  -  y     I  -  X      X  -  I     2x  +  z 
X  ? 


(a  -b){b  -  c)       (b  -a)(c-  b)       {a  -  b)  {c  -  b) 

15.  Express  f  >  |>  f  as  fractions  with  denominator  24. 

16.  Express  ->   yy   -  as  fractions  with  denominator  abc. 

a    b    c 

17.  Express  -7-'  wr'  t^  as  fractions  with  denominator  24a6c. 

4a  60  8c 

18.  Express  tt-^'  ^7^9  as  fractions  with  denominator  9aW. 

19.  Express  ^ ^>  x Sr  as  fractions  with  denominator 

2a  -  26  3a  +  3o 


6(a2  -  62). 


Arts.  9G,  97]          DIVISION  OF  FRACTIONS  149 

20.    Find  the  L.  CD.  of, J?^, tt  and  '^^ 


(«  -  by  (a  +  6)         (a  -  b)  {a  +  6)-' 

Reduce  to  a  single  fraction  and  simplify  (leaving  the  de- 
nominator factored)  : 

o.  5  3  or,  ^  ^ 

22.       ^  ^ 


2x- 

2U 

3x- 

31/ 

X 

T- 

X- 

X  + 

a:2-  1 

a 

1 

X  +  1      X*  +  4 

23  ^^        _         "  29 

36  -  2a      2a  -  36  a-  -  3a  -  4      a  -  4 

n.      ^    _    y  30  1 1 

X  -  2/      x  +  y  x2  -  4x  +  3     x2  -  3x  +  2 


c  c 


25.    r  +  ^ 31. 


X  -  1         1  +  X        x'^  +  1 

x  +  l~l-x~x'--l 


26.   ^±i^+^^.  32.   ^-^-       2      ^^ix 


33. 


X  -7j^  x  +  y  1  +  3x      1  -  3x  '  1  -  9x2 

1  1.1 


(z  -  x)  (x  -  y)      (x  -  y)  (y  -  z)      (y  -  z)  {z  -  x) 
97.    Division   of  fractions.     The   method  of  dividing  one 
fraction  by  another  is  the  same  in  algebra  as  in  arithmetic. 
By  the  definition  of  division,  the  divisor  times  the  quotient  gives 
the  dividend. 

Hence,  to  divide  §  by  f  is  to  find  a  number  q  such  that 
.3  2 

4 '^  =  3-  (1) 

Multiply  both  members  of  (1)  by  |.     This  gives 

2^4        8. 
«=3^3   =9 

In  general,  to  divide  t-  by  -.  means  to  find  a  number  q  such 

that 

c         a  (2) 

d'?  =  6' 


150  FRACTIONS  [Chap.  XVI. 

To  find  q,  multiply  both  members  of  (2)  by  -  •     This  gives 

a      d      ad  (3) 

^      h      c       he 

From  (3)  we  note  that  the  quotient  is  obtained  by  multiplying  the 
dividend  by  the  divisor  inverted. 

EXERCISES 

Perform  the  indicated  operations  and  simplify: 

1.    ^  -^    ^-  10.   ""      ^'      ' 


3  b  '  \d  '  fj 

12a  -  6      2a 


2.   ^\  ^  7-  11. 


2>x        '     Ibx^y 


„    4       4     5  .„       4-4a  +  a2  4-o2 


5    ■  3     9  aia"  -  a  -  12)   '  W  +  a^ 

ba^b       4c2  x^  +  y^        x  +  y 

c      '  5a^  '   x'^  -  9y^  '   X  +  Sy 

9x^      7xy  X  +  y        x  +  y 

6y'^  '    4:a  '   x^  +  x^y  '      x'^ 

fSabV  ^  (Saby  x^  +  y^  ^  x^y  -  xhf  +  xy^ 

\2c  J    '    {2cy  '   x"^  -  y^  '    x^  +  x^y  +  xy^ 

a       c\    e  ab         fa  +  b          b 

b^dj  'J  ^TVh  ^  \b~  ^  ^TTfe 


fa      c\      e  x^  -  X  -  Q)      x^  -  2x 


b   '  d)      /  X?  -  X  -2       X-  +  X 

hn  +  n      m  -  n\      fm  +  n      m  —  n 
19.    I  + 


n      m  +  nj       \m  -  n      m  +  n 

-■-f)^[@-')(^)} 


22. 


23. 


Arts.  97,  9S]  COMPLEX  FILVCTIONS  151 

V  .c       x~J       \x-  +  X  +  IJ 

V  X  +  ijj       \        X  +  yj 

26.  ri  +  i+3n     ri     2    31 

|_xi/      a;2      ?/0j       \_x      ij      z] 

98.  Complex  fractions.  A  complex  fraction  is  an  indicated 
division  in  which  either  one  or  both  terms  are  fractions  or  con- 
tain fractions.  A  complex  fraction  can  be  reduced  to  a  simple 
fraction. 

3 

Example  1.     Reduce  |  to  a  simple  fraction. 


Solution:   Multiply  both  terms  by  10,  the  L.  C.  D.  of  the  numerator 
and  denominator.     Then 

I  ^  lOli  =  15 
I     10  •  I      u 

Example  2.     Reduce ^  to  a  simple  fraction. 

y  "3^ 

Solution:   Multiply  both  terms  by  .3xy,  the  L.  C.  D.  of  the  numerator 
and  denominator.     Then 


2^      3x.(2.^)       ^^^^^^^ 
X      2y  (x      2y\       3x'  -  2y^ 


152  FRACTIONS  [Chap.  XVI. 

EXERCISES 
Reduce  the  following  complex  fractions  to  simple  fractions: 


1.    21. 
3f 


10. 

4-  ^ 
+    4 


11. 


4.    TT-  12. 


2  +  i 
3i  +  2i 

4f +  7^ 

1 

a 
2  ■ 

a2 

1 
a  +  fe 

b 
a 

x  +  l 

1 

re 

y 

--1 

I  +  a 

1  +  i- 

0 

a      b 

b+a 

a      b 
b      a 


13. 


14. 


15. 


16. 


17. 


X        X 

2  +  3 

2      X 

x  +  S 

a^  -b^ 

a 

a-b 

2a 

m  -n 

m  +  71 

m  -  n 

m  +  71 

7n  -  n      m 

+  n 

m  -\-  n      TTi 

-  n 

m  —  n      m 

+  n 

m  +71      m 

-  71 

1 

-ri. 

.-2+3 

X 

X      x^ 

x^      y' 

y2                3.2 

y2 

i+»  +  i 

-  m 

/?^          1 

-  711 

1  +  /// 

771 

Art.  98]  MISCELLAxNEOUS  EXERCISES  153 

21.    1 


xo. 

i  +  . 

1 
+  2 

a 

a 

19. 

1 

b 

1  -6=^ 

20. 

6 
1 

-• 

1  + 

1+a 
24. 

2x  + 

5+5- 

2  3 

.r  +  3      X  +  2 

X  +  5 

x\  6 

3  o 


fl  -  b      g  +  fe      a^  -  b^ 


3x  + 


4x 


MISCELLANEOUS  EXERCISES 

XX 

1.  Which  is  the  greater,  ^  or  .^>  when  x  and  y  are  positive 

numbers? 

2.  Which  will  give  the  greater  quotient,  if  x  and  y  are  posi- 

x 
tive  numbers,  to  divide  a  positive  number  by  ^  or  to  divide 

X 

the  same  number  by  — ?     Explain. 

2x  X 

3.  Fill  the  blanks  in  the  following:  5 5 -^ — ^=(     )j 

ofl  —  O       CI  —  o 

8-2x4 


(    )■ 


xy{2  +  X-) 

4.  The  reciprocal  of  a  numl)er  x  is  -•  That  is,  the  recipro- 
cal of  a  number  is  1  divided  by  that  number.  What  is  the 
product  of  a  number  and  its  reciprocal? 


154  .  FRACTIONS  [Chap.  XVI. 

5.  P'rom  the  definition  of  reciprocal  in  Exercise  4,  give  the 
reciprocals  of  the  following  :   (a)  2;  (b) '-. ;  (c)  -;  (d)  ——r]  (e)  ^- 

6.  Show  that  to  divide  by  a  fraction,  we  need  simpl}^ 
multiply  by  its  reciprocal. 

Perform  indicated  operations  and  simplify  : 


7. 

3a             a 
a^  -4:  •  a -2 

8. 

4:X  +  2      2z  +  1 
3a       '       5a 

9. 

4mV3  ^  -T 

r~  -  rx 

n 

x^  -  a^       x^  +  ax 

{x  -  ay 


11. 


X  +  ^ 


12.    ^^'. 


T 


\x^  +  y 


3xy 


Vl+a  a     J   '   \\  +  a 


20. 

a2  +  &2 

— i —  -  2a 

6      a 


13. 

a-h           i 

^          a-h 

14. 

1  ('O,,           Ql\                     ^ 

7(^2/   '^2)   y_^ 

15. 

4          3 
a      a  -  1 

16. 

1 

-ri. 

a;  +  ?/  1  x-y 

17. 

x-y      X  +  y 

X  +  y      x-y 

x-y      X  +  y 

x^  -  if 

18. 

X 

x-y 

3x 

x^-y 

-)• 

x-y 

1 

-"Y 

Art.  98]  EXERCISES  155 

'''"       -  1 


m-  —  n- 

ax  —  bx  +  ay  —  by 
23.    r  • 

axy  —  bxy 

Q>xyz  +  Zxh  +  3j/^g 
2a:2z  +  ^xyz  +  2r'2' 

flx  -  6x  +  ex  +  «?/  -  by  +  cy 
axy  -  6a:?/  +  ca:?/ 

5xyg  -  15yz  +  10xz  -  SOz 
5xh  -  5xz  -  302 


27. 
28. 
29. 


4ax^  +  14ax  -  30a 
Qx^  +  2\x^  -  45a:  ' 

6a:3  ^  48 
3a;3  _  6a:2  +  12a:' 

6ax^  -  aa:^  -  12ax 
'oax?  -  llax^  +  12aa;' 


(m  -\-  n      m  -  n\    (m  -\-  n      m 
30.      + 


31 


-  n\    /m  4-  n 
\m  -  n  '  m  +  i?/    \m  -  n      m  +  n 

''    V        a:  ^  a:V  '  x'  +  x+ 1    x 


a:^  -  a:  -  6  .  x^  -  2a:  -  3  .  .T-'  -  2a:  +  1 
^^'   x^  -  X  -  2'  x""  +  x  -  2  '      a-2  -  9 

a:^  +  2a:  +  4    a:^  +  8  .      a-=^  -  4 
^**        a:  +  2         a:^  -  8    a:^  -  4  +  2a:' 
35    c'^-(a  +  6)V  fl  c2_(a_5)2 

oc  -  a-  -  ab    ((I  -  c)-  -  b-    be  -  nb  +  b^ 


156  FRACTIONS  [Chap.  XVI. 

«.         1  1  2 

36. + 


1    '  1,1 

X  -  -      X  +  -      \  -  - 


37.    -1±^   +'-^  +  '— 

1  X  X 


\  -{-  X        \  —  X        1+rc 
2m  +  n 


38.    ^^^±^ 


1 
39. 


m  +  n 

X  +  \  a;  +  4 


{x  -  4)  (2  -  x)    '   (x  -  1)  (2  -  x) 

^Q     1    _  r^ i_i 

*  m  +  1       \jn  -  1      m^  -  ij 

41  ^  -  ^  ^  «  +  y  _         (^7  -  «)'       . 

"  a  +  2/      a  -  a;       (a  -  x)  {a  -  y) 

.«  1  1  1 

42.  -, Tx  + 


(a  -  b)   '   (a  -  c)    '   (6  -  c) 

,,111111 

43.    + + + ■  + + 

X  -  y      X  -  z      y  -  z      y  —  x      z  -  x      z  —  y 

..2  3  4 

44.  ,    ,    ^    +  ,    .    ^    + 


45. 


1  +  2x      1  +  3a;   '   1  +  4x 
X  1 


(.r  -  6)  {x  -  4)       (x  -  2)  (4  -  0-)    '    12  -  8ar  +  x" 


46.  '-i  +  iL^+2. 

.r      x-      X-  +  .T 

1  ^ 

47.  -^'-,  +  -, 


^  -  ?/-       X-  -  xy  —  2y'^-      x-  +  ?>xy  -\-  2y- 


+  .T 


2(1  -  a)  1  -  a 


a-  +  3a  -f  2      a-'  +  4a  +  3      a^  +  5a  +  6 


Art.  98]  REVIEW  EXERCISES  157 

REVIEW  EXERCISES 

1.  Give  an  example  of  a  linear  equation  in  an  unknown  x;  in  an  un- 
known y;  in  an  unknown  t. 

2.  State  whether,  aoeording  to  the  definition  of  a  factor,  (a)  ^  is  a 

factor  of  4;  (b)  -  is  a  factor  of  x-  ;   (c)  mn  is  a  factor  of  mn.     Name  all  the 
factors  of  ax^  ;  of  12. 

Find  the  following  products. 

3.  («-3)(«+7).           7.    (x+0)2.  11.  (00  +  7)\ 

4.  (2x  ~  l)(2x  -  4).        8.    (0  -  x)2.  12.  («  ~h  -  c)\ 

5.  (2x  +  ^)(2x  -\).        9.    (40  +  5)2.  13.  {Ah  +  0<  +  S)^ 

6.  (4<  +  wf.                   10.    (70  +  1) (70  -  1).     14.  (3x  +  7)(4x  -  5). 

15.  Tell  which  of  the  following  are  rational  and  integral  with  respect 

tox:    3a=x';  «+-x;    +-;    5\/x  +  17;  lAy/ax  +  x^. 
X  a 

16.  Tell  which  of  the  following  are  not  trinomial  squares  and  give  rea- 
sons:  (a)  x^  +  2ox  -{■  a";  (b)  m^  ~  2mn  -  n^;  (c)  a'-  -ab+b";  (d)  i+Sb  +  \6b\ 

17.  Describe  the  way  in  which  the  terms  of  a  trinomial  square  arc 
made  up. 

Combine  each  of  the  following  into  a  single  fraction: 


18.1-?. 

a      a 

-i. 

- 

1 

24. 

^  _  1. 
rs      qr 

2+6 

19.   -  +  X. 

22,   «. 

c 
d 

25. 

a 

-"-.4- 

23.   5- 

.5 

26. 

y     X 

a 

+  5' 

Simplify  the 

f  olio  wing: 

27.    a-'', 
a 

30.    (a  +  1)-^- 

33. 

4a   36 
36'  8 

'^•ilr 

28.   a. ^. 

31.   U^. 

a      a 

34. 

fl  +4 
2 

-^2. 

37."'  +  ' 

X 

a- 

-=-  — 

X 

29.  |.l(te. 

32.   g  .  I2py. 

35. 

4 

1     1 

38. "  :^. 

ab 

158  FRACTIONS  [Chap.  XVI. 

39.  Find  the  remainder  when  the  sum  of  the  squares  of  a  and  b  i.s 
taken  from  the  square  of  the  sum  of  a  and  b. 

40.  Find  the  remainder  when  the  difference  of  the  cubes  of  a  and  b 
is  subtracted  from  the  cube  of  the  difference  between  a  and  b. 

Perform  the  following  additions  and  subtractions: 

2a  -1       _        g  -3 
*^-   o2  _  8a  -  9      a^  -  7a  -  18 


a 

b 

a' 

+  2ab 

a? 

+ 

4ab 

+  4/)2 

a  —  X 

b  +c 

42         ^  -  7         I  ^^  44 

6x2+5x-4"^  10x2  +  7x -6  '    a.T  +  6x  +  cj:       (a  +  6)=  -  c^ 

Find  two  solutions  for  each  of  the  following  equations: 

45.  x2  -  6x  =  27.  47.   Sj^  -  4x  =  0. 

46.  4x2  _  16  =  0.  48.   Sx^  +  7x  =  6. 
»iil--^         1-x         x-1      x-1 

49.  Add:, > r- -.      :j 

1-2/        y  -  1        2/-1      1-2/ 

50.  Add  ^  "^  ^ 


(6-o)(c-d)       (c-6)(d-c)      (a-fe)(6-c) 


CHAPTER  XVII 

FRACTIONAL   AND   LITERAL   EQUATIONS 

99.  Clearing  equations  of  fractions.  We  have  solved 
(Chapter  IX)  some  equations  in  which  fractions  are  involved. 
It  is  often  convenient  to  clear  such  an  equation  of  fractions  by 
multiplying  each  member  by  the  L.  C.  M.  of  the  denominators. 

To  illustrate,  solve  the  equation        3x     3(x  -  1)      99  ,^. 

2  "^        5         ~  10'  ^^^ 

Solution  : 

Multiply  each  member  by  10,  the  L.  C.  M.  of  2,  5,  and  10. 

This  gives  15j:  +  6(j  -  1)  =  99.  (2) 

Transposing  and  collecting,  21x  =  105,  (3) 


3  3(5  -  1)       15      12  _  99 

(^heck:  2"^+         5         ~2"''5~10" 


EXERCISES  AND   PROBLEMS 

Solve  the  following  equations  and  verify   the  results  l)y 
substitution : 

1.  ix-l-n-  AK3x  +  1).      5.    f (x  +  1)  +  5x  =  46. 

2.  t.  +  ^x  =  4f.  6.2.-1-1^  =  5. 
x_+^      2x±l_  l"^ 

^  ^  J    5x-4  _  2x  +  Q  ^  2 

4x  +  1       1  '4  7 

4.   ^-^'  _  ^  (X  +  5)  =  4. 

o    5x      1,3/        1\        7       ^ 
159 


160    FRACTIONAL  AND  LITERAL  EQUATIONS  [Chap.  XVII. 

11.  {x  +  i)(2x  +  1)  =  {x  +  f)(2x  -  3). 

12.  (x  -  ly  -(x  +  l)(x  -  f)  =  0. 

13.  (x  -  2)(x  -  3)  -  xix  -  ■%4)  =  5. 
,,  x-1      x-2      11  -  13x 

15.    (x  -  ^)(x  +  f)  -x2  +  5x  =  3. 

100.  Unknowns  in  the  denominator.  In  some  equations 
the  unknown  appears  in  a  denominator  or  in  both  numerator 
and  denominator.  In  these  equations  as  in  the  equations  of 
Art.  99,  the  first  step  in  the  solution  is  clearing  the  equation 
of  fractions  by  multiplying  each  member  by  the  L.  C.  M.  of 
the  denominators.  Why  the  L.  C.  M.  is  used  rather  than  any 
common  multiple  of  the  denominators,  will  be  taken  up  in  the 
second  course. 

Example.     Solve  2a:  -  1    _  3(x  -  2) 

2(x  -  3)  ~    3a;  -  1  ■ 

Solution  :  The  L.  C.  M.  of  the  denominators  is  2 (a;  -  3)  (3a:  -  1). 
Multiplying  both  members  of  the  equation  by  this  expression  we  obtain 

{2x  -  l)(3a:  -1)  =  6(a:  -  3)(a:  -2), 
or  6.r;2  -  5a;  +  1  =  Gx^  -  30x  +  36. 

Then  25x  =  35, 

and  X  =  -^■ 

5  • 

7  .  .  . 

Substituting  x  =  -  in  the  original  equation,  we  find  the  equation  satisfied. 

EXERCISES 

10      491  a:  +  la:-5 


3. 


a;9a;2  x  +  3      x 

X  -  8      X  -  5 
X  —  9      X  —  7 


5  3 

4.   — ^  + 


X  +  3   '   2  (a;  +  3)       2      2  (a-  +  3) 


Art.   100]  EXERCISES  AND  PKOliLEMS  161 

,71       23  -  .r       7         1 

5.    — —  =  -I . 

a;  -  1    ^  a-  +  1      x^  -  \ 

^     13  (a: +  4)  _  3(2x-l)  ^  ^ 
X  +  5  X  +  1  '■ 


10. 


a;  -  4      .r  -  3      .r  -  1      a:  -  5 

_± 1_  ^_4_ 1_ 

X  +  3      .r  +  5      .r  +  2      .r  +  1 ' 

_4 3_^      3  4 

1  +a:      3  +x      1  -  a-      2  -  .r" 


x-l      .r-3      X  -  o      X  - 
6  9  4-1 


12. 
13. 


x-3      a:-2      a:-l      a;-4 

_^ 3  4  11 

.T  -  1      .r  +  2      7(a:  -  3)      7(.t  +  4)' 


PROBLEMS 

1.  One-half   of  a  certain  number  plus  one-third  of  the 
number  is  10.     Find  the  number. 

2.  The  sum  of  two  numbers  is  66.     One-half  of  the  smaller 
plus  one-seventh  of  the  greater  is  18.     Find  the  numbers. 

3.  The  difference  between  one-third  of  a  number  and  one- 
fifth  of  the  number  is  6.     Find  the  number. 

4.  The  sum  of  a  certain  number,  its  half,  its  tiiinl,  its 
fourth,  and  its  fifth  is  274.     What  is  the  numl)er? 

5.  What  number  must  be  added   to   the  numerator  of 
/y  in  order  that  the  resulting  fraction  shall  be  equal  to  i? 

6.  What  number  must  be  added  to  the  denominator  of 
#r  i'l  order  that  the  resulting  fraction  shall  be  equal  to  ;? 


162    FRACTIONAL  AND  LITERAL  EQUATIONS  [Chap.  XVII. 

7.  What  number  must  be  added  to  both  numerator  and 
denominator  of  -^j  in  order  that  the  resulting  fraction  shall 
be  equal  to  f  ? 

8.  A  yardstick  is  cut  into  two  pieces.  One  piece  is  § 
the  length  of  the  other.     What  is  the  length  of  each  piece? 

9.  What  number  added  to  both  numerator,  and  denom- 
inator of  the  fraction  f  will  double  the  value  of  the  fraction? 
What  number  will  halve  the  value  of  the  fraction? 

10.  Find  two  numbers  which  differ  by  4,  and  such  that 
one-half  the  greater  exceeds  one-sixth  of  the  lesser  by  8. 

11.  $1000  is  divided  between  A  and  B  in  the  proportion 
3  to  8.     How  much  more  is  B's  share  than  ^'s? 

12.  A  man  leaves  one-third  of  his  property  to  his  wife,  one- 
fifth  to  each  of  his  three  children,  and  the  remainder,  which  was 
$1200,  to  other  relatives.     What  was  the  value  of  his  estate? 

13.  An  estate  of  $5300  was  left  to  two  heirs.  The  first 
received  one-third  more  than  the  second  and  $400  additional. 
How  was  the  estate  divided? 

14.  A  man  has  two  sums  of  money  at  interest,  which  to- 
gether amount  to  $19,000.  One  sum  brings  5%  interest,  the 
other  3%.  From  the  latter  he  receives  $250  more  income  than 
from  the  former.     What  is  the  amount  of  each  of  the  two  sums? 

15.  The  width  of  a  room  is  two-thirds  of  its  length.  If 
the  width  were  five  feet  more  and  the  length  8  feet  less  the 
room  would  be  square.     What  are  the  dimensions  of  the  floor? 

16.  Each  year  a  merchant  increases  his  capital  one-third, 
but  takes  away  $4000  for  expenses.  At  the  end  of  the  second 
year,  after  deducting  the  .second  $4000  he  finds  that  his  capital 
has  increased  one-half.  What  was  his  capital  when  he  began 
business? 

17.  The  denominator  of  a  fraction  exceeds  its  numerator 
by  2.  If  1  is  added  to  both  numerator  and  denominator  the 
resulting  fraction  will  be  equal  to  f .     What  is  the  fraction? 

18.  A  lazy  boy  is  asked  to  divide  one-half  of  a  certain 
number  by  8,  and  the  other  half  by  10,  and  add  the  quotients. 


Art.  100]  PROBLEMS  163 

To  save  work,  ho  tlivitleil  the  nuinljcr  itself  by  9,  and  obtained 
a  result  too  small  by  one.     Determine  the  number. 

19.  At  the  time  of  their  marriage  a  man's  age  was  to  that 
of  his  wife  as  3  is  to  2.  Nine  years  later,  it  was  as  4  is  to  3. 
What  was  the  age  of  each  at  the  time  of  their  marriage? 

Hint:   Let  3x  =  age  of  the  man  at  marriage. 

20.  A  tank  is  emptied  by  two  pipes.  One  can  empty  it  in 
30  minutes,  the  other  in  25  minutes.  In  what  time  can  the 
two  together  empty  it? 

21.  A  can  do  a  piece  of  work  in  3  days,  and  /i  can  do  it  in 
5  days.     In  what  time  can  tliey  do  it  together? 

Hint:    Let  x  =  the  number  of  daya  it  will  take  A  and  B  together,  then 

-  =  the  part  .d  and  B  can  do  in  one  day. 

X 

22.  If  A  can  do  a  piece  of  work  in  8  days  and  B  in  10  days, 
in  what  time  can  they  do  it  together? 

23.  Smith  and  Jones  have  forgotten  the  scores  in  the  foot- 
ball game  between  Chicago  and  Illinois  in  the  fall  of  1914. 
Jones  remembers  that  the  difference  of  the  scores  was  14,  and 
Smith  remembers  that  the  difference  of  the  scores  was  half  as 
much  as  the  sum  of  the  scores.  What  were  the  scores  of  the 
two  teams? 

24.  Smith  and  Jones  have  forgotten  the  scores  of  the  foot- 
l)all  game  between  Chicago  and  Illinois  in  the  fall  of  1913. 
Smith  remembers  that  Chicago  won  by  16  points,  and  Jones 
remembers  that  Illinois's  score  was  2  less  than  half  that  of 
Chicago.     What  were  the  scores? 

25.  A  man  is  now  45  years  old  and  his  son  is  15.  How 
many  years  must  elapse  before  their  ages  will  be  as  13  is  to  7? 

26.  Find  a  fraction  whose  numerator  is  greater  by  3  than 
half  of  its  denominator,  and  which  when  reduced  to  its  lowest 
terms  is  §. 

27.  Find  the  age  at  which  the  Greek  mathematician  Dio- 
phantus   died.     His   epitaph   reads   as   follows :     Diophantus 


164    FRACTIONAL  AND  LITERAL  EQUATIONS  [Chap.  XVII. 

passed  |  of  his  life  in  childhood  ;  j\  in  youth  and  |  more  as  a 
bachelor  ;  five  years  after  his  marriage  was  born  a  son  who 
died  four  years  before  his  father  at  half  his  father's  age.* 

101.  Literal  equations.  A  literal  equation  is  one  in  which 
some  or  all  of  the  known  numbers  are  represented  by  letters. 
It  has  become  customary  to  represent  the  unknowns  by  the 
last  letters  of  the  alphabet,  while  the  known  numbers  are 
usually  represented  by  the  first  letters. 

Thus,  the  general  hnear  equation  (Art.  65) 
ax  +b  =  0 
is  a  literal  equation.     Other  examples  of  -literal  equations  are 

-  +b  =  c,     {a  -b)x  =-b  +c,     (3  -  a)  +  (2  +  6)x  =  c. 

EXERCISES 

Solve  the  following  equations,  the  unknowns  being  repre- 
sented by  the  last  letters  of  the  alphabet : 

1.   -  +  6  =  c. 

X 

Solution:  Multiply  both  members  by  x,  and  obtain 

a  +bx  =  ex.  (1) 

Transposing  and  collecting  terms,  we  have 

(6  -  c)x  =  -a, 

,  —  a  a  .  - . 

and  X  =  ^ — ; ,  or  - — j^  ■  (2) 

Check:  Substitute  in  (1), 


^'b-c'^ 

c 

-b 

ba           ca 
+  c  ~b~c  -b 

ab  +  ah  =  ac. 

0=0. 

*-i  + 

h  _ 
X 

=  c. 

2.  (a  -  h)x  =  b  +  c. 

3.  (3  -  a)  +  (2  +  h)x  =  c.  5.   ay  +  cy  =  2(a  +  h). 


*  Father's  age  here  means  his  age  at  his  death,  and  not  at  time  of  his 
's  death. 


Ahts.  101,  102]          SUBSCRIPT 

NOTATION 

6.   az  +  bz  =  3(rt  +  b). 

10.   a.s  -  a^  -  4  =  3a 

7.   ex  +  dx  =  c^  -  dK 

11    a          ^ 

c  +  dx 

«    o6      ,  ,             1 
8.   —  =  6c?  +  c  +  - . 
u                       u 

12.    ,,.'•("+1). 

9.   a(2a  +  x  -  1  +  d)  = 

26. 

13.   i  =  i  +  Ul 

w      a       b      c 

165 


14.    (a  +  x)(6  -  x)  -  (a  -  x){b  +  x)  =  2. 

M^  ~  6^  ^  ^^TTp- 

16    ^  +  "  _      ^     ^  1  _| ^  . 

r  r-a       r       r-a 

a  —  b      at  -  b      by  -  a      ^  ,      . 

17.  a  H TT-  =  — TT—  H -. —    Solve  for  ij  and  then  for  t. 

abt  bt  at 

X  XX 

,.    "  +  2  ,,    "  +  2      "+3 

18.   =  a.  19.   = 

X  XX 


20.    =  2  + 


„  +  26  -  a6 

+  6 


102.  Subscript  notation.  It  is  often  convenient  to  repre- 
sent related  nunil)ers  by  the  same  letter  with  small  numbers 
(called  subscripts)  written  at  the  right  and  below  the  letter. 
Thus,  if  T  represents  temperature,  the  temi)erature  of  a  body 
at  two  different  times  may  be  representetl  by  Ti,  T^.  The 
weights  of  three  bodies  may  be  represented  by  H'l,  IT..,  H'a. 
These  letters  are  read  "  T  sub  1,"  "  T  sub  2,"  "  11'  sub  1,"  and 
so  on. 

Another  notation  less  often  used  is  tiie  prime  notation. 
Two  or  more  letters  are  distinguished  from   one  another  by 


166    FRACTIONAL  AND  LITERAL  EQUATIONS  [Chap.  XVII. 

marks  (called  primes)  placed  at  the  upper  right  hand.  Thus, 
temperatures  at  two  times  may  be  represented  by  T',  T",  and 
weights  of  different  bodies  by  W,  W",  and  W".  These  are 
read  "  T  prime,"  "  T  second,"  "  W  prime,"  "  W  second,"  and 
"  W  third." 

Both  the  subscript  and  prime  notation  are  used  in  physical 
and  engineering  formulas. 

TRANSFORMATION   OF   PHYSICAL  FORMULAS  AND  PROBLEMS 

In  the  following  physical  formulas,  solve  for  the  letter 
indicated.  The  custom  of  using  the  first  and  last  letters  of 
the  alphabet  for  the  knowns  and  unknowns  respectively  is  not 
followed  here. 

1.  h  = ;•     Solve  for  i. 

1  +  I 

2.  H  =  1082  +  .305^.     Solve  for  t. 

3.  C  =  |(F  -  32).     Solve  for  F. 

4.  C  =  -jz Solve  for  n. 

R  +  nr 

5.  C  =  -f: — -. Solve  for  r. 

R  +  nr 

6-  Q  =  „/    t         •     Solve  for  R. 

W  +  R  +  w 

^    ,„„      W'(H  -t  +  32)      ^  ,      „     , 

7.    II     =  —     „„^ •     Solve  for  t. 

966 

8-    VJi  =  L  +  ^/     Solve  for  d  and  for  t. 

at 

W  -  w 
9.   A  =  II  YTr TIF, TT^r     Solve  for  IT. 

10.   F  =  (-A-^  (-^\    Solve  for  r'. 


Art.    102] 


EXERCISES   AND    I'KOHLEMS 


1G7 


Read  aloud  the  following  fornuilas,  then  solve  for  the  letter 
indicated. 

11.  Cih  -  to  +  W,{(,  -  /,)  =  ir2(/o  -  t,).     Solve  for  ^3. 

12.  ]\  =  ]'o(l  +  .0037/,).     Solve  for  tu 


13.    A' 


ri(/-2  -  U) 


Solve  for  r,. 


14.    ((?i  -  Qo)  =  lll_^^i.     Solve  for  T,. 
i  1 

^_     jf      7.65       765      „  ,      .      ^„ 
•15.   H  =  — ^  -  y7^-     Solve  for  T". 

16.   ^  =  i6/i"  +  ^a(h"  +  h')  +  \ch'.     Solve  for  /i' 


17.    P"  = 


P' 

V2 

2 

V-i\ 

Solve  for  ?J2. 


18.  If  a  and  6  are  the  altitude  and  base  of  a  triangle  and  ^4 
the  area,  then  A  =  —•     Express  a  in  terms  of  b  and  A. 

19.  If  r  per  cent  is  gained  on  an  article  that  costs  c  dollars, 


the  selling  price  S  is  given  by  the  formula  »S'  =  c  + 


100 


Find  c  if  r  =  10  and  the  selling  price  is  .S181.50. 

20.  Write  a  formula  for  the  selling  price  when  r  per  cent  is 
lost.  Find  r  when  the  selling  price  of  the  article  in  Problem 
19  is  $138.60. 

21.  The  proceeds  P  of  an  amount  of  a  dollars  discounted 

art 


for  t  days  at  r  per  cent  a  day  is  P  =  a 


36000 


Solve  the 


equation  for  a  ;  for  r  ;  for  /. 

22.    If  a  cents  be  divided  between  two  Imys  .so  that  one 
boy  has  6  cents  more  than  the  other,  how  many  cents  has  each? 


168    FRACTIONAL  AND  LITERAL  EQUATIONS  [Chap.  XVII. 

23.  The  pressure  of  water  per  square  inch  at  a  depth  d  feet 

62.5 

is  given  l)y  the  formula  P  =  -r^  d.     At    what    depth  is  the 

pressure  ten  times  as  great  as  the  pressure  of  the  air  at  the 
surface?  (Pressure  of  the  air  to  be  taken  as  15  pounds  per 
square  inch.) 

24.  If  two  quantities  of  water  nii  and  m2  at  temperatures 
ti  and  t2  are  mixed,  the  temperature,  t,  of  the  mixture  is 

_  mih  +  m2t2 
w?i  +  7n2 

How  much  water  of  temperature  48°  must  be  mixed  with  6 
gallons  at  150°  to  make  a  mixture  at  110°? 


CHAPTER    XVIII 
RATIO,   PROPORTION,   AND   VARIATION 

103.  Ratio.  The  ratio  of  a  number  a  to  a  sceond  number 
6  is  the  quotient  obtained  by  dividing  a  by  b.     The  ratio  of  a 

to  b  is  usually  written  in  the  fractional  form,  r'  but  the  form 

b 

a  :  6  is  often  found.     AH  ratios  of  numbers  may  be  considered 

as  fractions. 

The  two  numbers  a  and  b  in  a  ratio  are  called  the  terms 

of  the  ratio.     The  numerator  is  called  the  antecedent  and  the 

denominator  the  consequent. 

EXERCISES 

In  the  following  exercises  name  the  antecedent  and  the 
consequent  for  each  ratio;  write  the  ratio  in  fractional  form 
and  simplify  by  reducing  the  fraction  to  its  lowest  terms: 

1.  Ratio  of  4  to  12. 

2.  Ratio  of  12  to  4. 

3.  Ratio  of  y\  to  6. 

4.  Ratio  of  7|  to  f\. 

5.  Ratio  of  3  minutes  to  2  minutes. 

6.  Ratio  of  3  minutes  to  2  seconds. 

7.  Ratio  of  X-  to  xy. 

8.  Ratio  of  a^  -  b-  to  a  +  b. 

9.  Ratio  of  16  pounds  to  26  pounds. 

10.  Ratio  of  16  pounds  to  26  ounces. 

11.  OfryV- 


12.   xjjz 


im 


170  RATIO,   PROPORTION,   VARIATION       [Chap.  XVIII. 


104.   Proportion.     A  proportion  is  an  expressed  equality  of 
)  ratios, 
proportion  is 


two  ratios.     If  the  two  equal  ratios  are  j-  and  -^7  then  the 


a      c 

and  a,  b,  c,  d  are  said  to  be  in  proportion.     The  proportion  is 
often  written 

a:b  =  c:d. 

An  older  notation  sometimes  found  is    a:b  ::  c:  d.     In    any 
notation  the  proportion  may  be  read  "a  is  to  6  as  c  is  to  d." 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms  the  means.  Thus, 
in  the  proportion, 

a  _  c 

b~d' 

a  and  d  are  the  extremes  and  b  and  c  the  means. 

Since  a  proportion  is  an  equality,  operations  that  may  be 
performed  upon  the  two  members  of  an  equality  may  be  per- 
formed upon  the  two  ratios  of  a  proportion.  Thus,  in  the 
proportion, 

a      c 

b^d' 

both  members  may  be  multiplied  by  bd,  giving 

ad  =  be, 
which  proves  the  theorem: 

In  any  proportion  the  product  of  the  means  equals  the  product 
of  the  extremes. 


Arts.  104, 105,  lOG]     MEAN  PKOPUHTIOXAL  171 

EXERCISES 
In  the  following  proportions  tost  the  above  theorem: 

1    ?  =  L^  „    17      53^ 

'   5      35  "*•   13  "404" 

„    11      143  ^     5x        85x2 

4. 


7        91  •    I7y      289xy 

6. 


a 
ax  +  ay  _      b 
bx  -  by      X  -  y 
X  +  2/ 


Find  the  value  of  .r  in  the  following  proportions : 
»     2      7 


3      X 


11.   a  :b  =  X  \c. 


9      14 

7.  -  =  — •  12.   a:x  =  c:  d. 
X       3 

8.  X  :  3i  =  3  :  8.  13.   3  :  x  -  2  =  7  :  x  +  3. 

9.  -  :  3^  =  3  :  8.  14.    -2  :  x  -  4  =  3  :  x  +  2. 

X 

10.    a:b  =  c:x.  15.    .r  :  40  =  100  tt  :  4  tt. 

105.  Mean  proportional.  If  the  means  of  a  proportion 
are  the  same  number,  this  number  is  called  a  mean  propor- 
tional between  the  two  extremes.  Thus,  in  the  proportion, 
2  :  G  =  6  :  18,  6  is  a  mean  proportional  between  2  and  18.  In 
the  proportion  n  -.x  =  x  :b,  x  is  a  mean  proportional  bet  ween 
a  and  b,  and  from  this  we  obtain 

X  =  ±  V^. 

106.  Third  and  fourth  proportional.  A  third  propor- 
tional to  two  numbers,  a  and  b,  is  the  number  .r,  such  that 

a  _b 
b~  x 


172  RATIO,   PROPORTION,   VARIATION       [Chap.  XVIII. 

Thus  in  fV  =  |-f ,  48  is  a  third  proportional  to  3  and  12. 
A  fourth  proportional  to  three  numbers  a,  b,  and  c  taken 
in  the  order  given  is  the  number  x  if 

a      c 
b      X 

Thus  in  f  =  Vj  9  is  a  fourth  proportional  to  5,  3  and  15. 


EXERCISES 

Find  the  mean  proportionals  between  the  following  pairs 
of  numbers: 


1. 

2  and  8. 

5. 

1       J    1 
5  ^"^20- 

2. 

4  and  9. 

6. 

x^  and  if. 

3. 

2  and  32. 

7. 

4        ,   16 
-„  and  — „ 

X^             Xlf 

4. 

1  and  64. 

8. 

-  and  z. 

z 

Find  third  proportionals  to  the  following  pairs  of  numbers  : 
9.   2  and  4.  12.    (a  +  b)  and  (a  -  b). 

10,  ^  and  =•  13.    -  and  — 
3          7  X  y 

11.  land  -1.  14.   -^^^^and^. 
7  X  -  7j  y 

Find  fourth  proportionals  to  the  following  sets  of  numbers 
in  the  order  given : 

15.  1,  2,  3.  18.    c,  b,  a. 

16.  3,  2,  1.  19.    a  +  b,  (lb,  a  -  b. 

17.  n,  b,  c.  20.   ->  ~.   -• 

X    y    z 


Arts.  107,  lOS,  109]  PROPORTION  173 

107.  Proportion  by  alternation.     If,  in  the  proportion 

a  _  c 
b  ~  d' 

both  members  of  the  equation  are  muhipUed  by  --  tiie  equation 

])ecomes  after  re(hietion 

a  ^b 
c      d 

That  is,  in  a  proportion  the  means  may  be  interehanged  with- 
out destroying  the  equahty.  In  this  case  the  second  propor- 
tion is  said  to  be  obtained  from  the  first  by  alternation. 

108.  Proportion  by  inversion.     From  the  proportion, 

a  _  c 
b~~d' 
we  have  from  Art.  104,  (1) 

be  =  ad. 


Dividing  both  members  of  this  equahty  by  ac,  we  have 
b      d 


(2) 


It  may  be  noted  that  the  members  of  (2)  are  obtained  from  (1) 
by  inverting  the  fractions.  For  this  reason  the  second  propor- 
tion is  said  to  be  obtained  from  the  first  by  inversion. 

109.    Proportion  by  composition.     If,  in  the  i)roi)ortion, 

1  be  added  to  both  members,  we  find 

fl       ,       c       ,                           a  +  b      c  +  d  .  . 

-6  +  l=rf  +  ''  "'  -h rf-  <2) 

In  this  case,  the  second  proportion  is  said  to  be  obtained  from 
the  first  by  composition. 


174  RATIO,  PROPORTION,   VARIATION      [Chap.  XVIII. 

110.  Proportion  by  division.  If  1  be  subtracted  from  both 
members  of  the  proportion, 

a      c 

there  results  — r —  =  — -. — > 

h  d 

and  the  second  proportion  is  said  to  be  obtained    from  the 

first  by  division. 

111.  Proportion  by  composition  and  division.  From  the 
proportions  by  composition  and  by  division  we  obtain,  after 
dividing  members, 

a  +  6  _  c  +  f/ 
a  -  h      c  —  d 

This  proportion  is  said  to  be   obtained  from  the  proportion 

7  =  -    by  composition  and  division. 

Composition,  division,  and  composition  and  division  are  often  very 
appropriately  called  addition,  substraction,  and  addition  and  substraction 
respectively. 

EXERCISES 

Write  proportions  obtained  from  the  following  by  (a) 
alternation,  (b)  inversion,  (c)  composition,  (d)  division,  (e) 
composition  and  division: 

1     ^  =  1  q    ?=21 

4      12  7      49 

9       10 
2.5=25-  4.3:11=8:41. 

5.   3  :  4  =  (.r  +  y)  :  {x  -  y). 

From  each  of  the  following  equated  products  obtain  five 
proportions : 

2    4      2    4 
6.2.10=4.5.  .       8.  3-7=7--3- 

7.   7.3-21.1.  9.   2-(x+y)=i-(x-y). 


Arts.  Ill,  112,  113]      VARIABLES  AND  CONSTANTS 

10.  3-  4  =  X-  1.  12.   *fl, -oo  =  biho. 

11.  2-3=6.  13.    Ti-  T2=  xi-  x-i. 

14.    If  -  =  — ,  prove 


15, 
16.    If 


y 

Z                     X               w 

Ol 
02 

h\              fli  —  02      hi  —  62 

=  T-,  prove =  -- r--- 

62                   oi               6i 

Xi 
X2 

^  prove  ^^  +  ^^^-^^+^^'' 

y2                               X2                           2/2 

Find  the  value  of  x  in  order  that  the  following  proportions 
hold: 

17.    (x  +  1)  :  (x  -  1)  =  2  :  3.       19.    (x  +  2)  :  (3x  +  4)  =5:6. 

2x  +  3      3 


18.    (x  +  1)  :  (x  +  2)  =  2  :  3.      20. 


3x  +  2      2 


112.  Variables  and  constants.  If  t  represents  the  time 
since  a  Chicago  train  left  New  York  and  s  its  distance  from 
New  York,  we  note  that  each  of  these  symbols  s  and  t  takes 
different  values  during  the  progress  of  the  train.  On  this 
account,  t  and  s  are  said  to  be  variables.  If  the  train  should 
run  at  a  uniform  speed  of  v  miles  per  hour,  we  know  that 

s  =  vt. 

In  general,  a  symbol  is  said  to  be  a  variable  if  it  may  repre- 
sent different  numbers  in  a  discussion  or  problem.  It  is  con- 
stant if  it  represents  only  one  number.  The  idea  of  a  variable 
l)lays  an  important  role  in  algebra  (Chap.  XIX). 

113.  Direct  variation.  When  two  variables  are  so  related 
that  their  ratio  is  a  constant,  either  one  is  said  to  vary  as  or  to 
vary  directly  as  the  other. 

In  symbols,  -  =  k  or  y  =  kx, 

when  fc  is  a  constant,  may  be  read  "j/  varies  as  x." 
*  See  Art.  102. 


176  RATIO,   PROPORTION,   VARIATION      [Chap.  XVIII. 

In  describing  the  progress  of  the  train,  (Art.  112)  we  write 


and  say  that  the  distance  from  New  York  varies  as  the  time 
since  leaving  New  York.     Suppose  i;  =  40  miles  per  hour,  then 

s  =  40^. 

EXERCISES 

Write  in  the  form  of  an  equation  each  of  the  following 
statements  : 

1.  The  weight  W  of  the  water  in  a  tank  varies  as  the 
volume  V  of  the  water. 

2.  The  simple  interest  earned  on  a  principal  P  varies  as 
the  time  t. 

3.  The  speed  «;  of  a  falling  body,  started  from  rest,  varies 
as  the  time  t  since  it  began  to  fall. 

4.  If  ij  varies  as  x,  and  x  =  2  when  y  =  6,  find  y  when 
X  =  10. 

Solution:   The  statement  y  varies  as  x  means 

y  =  kx     {k  constant).  (1) 

To  determine  k,  make  x  =  2,  ?/  =  6. 

This  gives  6  =  2k,  (2) 

or  ^  =  3.  (3) 

From  (I)  and  (3),  y  =  Sx.  (4) 

Substituting,  y  =  30. 

5.  If  y  varies  as  x,  and  a;  =  4  when  y  =  12,  find  y  when 
X  =  10. 

6.  The  area  of  a  circle  varies  as  the  square  of  its  radius. 
If  a  circle  whose  radius  is  10  feet  contains  314.16  square  feet, 
find  the  area  of  a  circle  whose  radius  is  8  feet. 

7.  In  using  a  spring  balance,  the  principle  apphed  is  that 
the  stretch  s  of  the  spring  varies  as  the  weight  W  to  be  de- 
termined. A  weight  of  10  pounds  stretches  a  certain  spring 
I  inch,  how  great  a  weight  is  required  to  stretch  it  2j  inches? 


Art.  113]  EXERCISES  AND  PROBLEMS  177 

8.  The  distance  through  which  a  body  fails  from  rest 
varies  as  the  square  of  the  time  in  seconds.  If  a  body  falls 
16  feet  the  first  second,  how  far  does  it  fall  in  10  seconds? 

9.  The  weight  of  a  wire  varies  as  its  length.  It  is  found 
that  100  feet  of  a  certain  wire  weighs  3  pounds.  Find  the 
weight  of  one  mile  (5280  feet)  of  this  wire. 

EXERCISES   AND    PROBLEMS 

1.  Find  a  mean  proportional  between  3  and  48. 

2.  Find  a  fourth  proportional  to  5,  6,  and  35. 

3.  Find  x  in  the  proportion  50  :  75  =  90  :  x. 

4.  In  a  certain  business  transaction,  A  gains  S200  and  B 
loses  S50,  and  then  A's  capital  is  to  B's  capital  as  4  to  1.  If 
A's  original  capital  was  S1200,  what  was  B's? 

5.  Find  two  numbers,  one  being  twice  the  other,  such  that 
their  sum  is  to  their  difference  as  5  : 1. 

6.  How  may  $10  be  divided  among  three  boys  so  that 
for  every  dollar  the  first  receives,  the  second  shall  receive  15 
cents  and  the  third  10  cents? 

7.  The  sides  of  a  rectangle  are  in  the  ratio  2  to  3.  Find 
the  ratio  of  the  area  of  a  square  of  the  same  perimeter  to  the 
area  of  this  rectangle. 

8.  In  the  state  of  Illinois  in  the  1910  census,  the  ratio  of 
foreign-born  white  inhabitants  to  native-born  white  was  ap- 
proximately 27.8  per  cent.  What  number  of  each  if  the  total 
white  population  is  5,527,000? 

9.  A  business  worth  .'§37,000  is  owned  by  three  men.  The 
share  of  the  one  man  is  a  mean  jiroportional  between  the  shares 
of  the  other  two.  If  his  share  is  S12,000,  what  is  the  share  of 
each  of  the  other  two? 

10.  Prove  that  no  four  consecutive  integers  such  as   //, 
n  +  I,  n  +2,  n  +  3  can  form  a  proportion. 

11.  The  1st,  3rd,  and  4th  terms  of  a  proportion  are  x  +  y, 
X  -  y,  and  (x  +  yY  ;  required  the  2nd  term. 


178  RATIO,   PROPORTION,   VARIATION      [Chap.  XVIII. 

12.  If  four  numbers  x,  y,  z,  and  iv,  not  all  equal,  are  in 
proportion,  show  that  no  number  different  from  zero  can  be 
added  to  each  which  will  leave  the  resulting  four  numbers  in 
proportion. 

13.  If  ax  -  by  =  cy  -  dx,  find  the  ratio  oi  x  to  y  in  terms 
of  a,  h,  c,  and  d. 

14.  Divide  44  into  two  parts  such  that  the  less,  increased 
by  one,  shall  be  to  the  greater,  decreased  by  one,  as  5  is  to  6. 

APPLICATIONS   TO   PROBLEMS  FROM   MENSURATION 

Two  triangles  that  have  the  same  shape  are  said  to  be  similar.  In 
two  similar  triangles  the  sides  of  the  one  taken  in  any  order  are  proportional 
to  the  sides  of  the  other  taken  in  the  same  order. 

15.  The  sides  of  a  triangle  are  12,  15,  and  20.  In  a  similar 
triangle,  the  side  corresponding  to 
12  is  18.     Find  the  other  sides. 


Hint:     Let  \bx  =  one  required  side 
and  20x  the  other. 

16.  The  sides  of  a  triangle  are 
12,  15,  and  20.  The  perimeter  of  a  similar  triangle  is  188  feet. 
Find  its  sides. 

The  areas  of  similar  triangles  are  in  the  same  ratio  as  the  squares  of 
the  lengths  of  corresponding  sides. 

17.  A  triangular  field  has  sides  30,  40,  50  rods.  Find  the 
sides  of  a  similar  field  of  four  times  the  area. 

18.  A  triangular  field  has  an  area  of  20  acres.  What  is 
the  area  of  a  similar  field  with  twice  the  perimeter? 

19.  A  triangular  lot  has  one  side  10  rods  and  has  an  area 
of  40  square  rods.  What  is  the  area  of  a  similar  lot  whose 
corresponding  side  is  25  rods? 

20.  The  areas  of  two  similar  triangles  are  121  and  144, 
respectively.  One  side  of  the  one  triangle  is  22.  Find  the 
corresponding  side  of  the  other. 


AiiT.   113]  EXERClSi;S  AND  PROBLEMS  179 

21.  A  horse  tied  with  a  rope  40  feet  long  in  the  center  of  a 
pasture  eats  ail  the  grass  within  reach  in  4  days.  If  the  rope 
were  20  feet  longer,  how  many  days  would  it  take  him  to  eat 
all  the  grass  within  reach? 

Hint:     The  area  of  a  circle  varies  as  the  square  of  the  radius. 

22.  The  volume  V  of  a  sphere  varies  as  the  cube  of  the 
radius  r,  and  the  volume  of  a  sphere  of  radius  10  inches  is  4189 
cubic  inches.     Find  the  volume  of  a  sphere  of  radius  8  inches. 

EXERCISES   IN   THE   NOTATION   OF  PHYSICS* 

23.  The  ratio  of  the  force  F  to  mass  rn  is  a  number  k. 
Express  each  of  the  letters  F,  m,  and  k  in  terms  of  the  other 
letters. 

24.  The  product  of  the  pressure  p  by  the  volume  v  of  a 
gas  divided  by  its  temperature  T  is  a  number  k.  Expre.<:s  each 
of  the  letters  p,  v,  and  T  in  terms  of  other  letters. 

25.  Given  7-,  =  ^7— >  solve  for  R,  h,  and  r  in  turn. 

li      27rr 

26.  Given  7-,  =  — ^3:; — .  solve  for  G. 

„„    ^.  V-x         R-h      ,      . 

27.  Given  -77 =         „  >  solve  for  x. 

V  -  X  +  V  P 

28.  Suppose  V  varies  as  /  and  v  =  128  when  t  =  4,  find  r 
when  t  =  25. 

29.  Given  that  s  varies  directly  as  the  square  of  t  and  that 
s  =  64  when  t  =  2.  Write  this  in  the  form  of  an  eciuatioii,  and 
find  s  when  t  =  10. 

30.  The  force  F  acting  on  a  body  varies  as  the  acceleration 
a  produced  in  the  motion.  Write  the  relation  between  force 
and  acceleration  when  the  constant  ratio  is  the  mass  m. 


*  The  teacher  is  not  ex|H'cto(l  to  take  the  time  to  cxijlain  the  physical 
meaning  of  the  rehitions  sivcn.  The  exerci.ses  an>  nivcii  to  faiiiiliarizf  the 
Ktudent  with  the  use  of  other  letters  than  x  and  y  for  unknowns. 


CHAPTER  XIX 

GRAPHICAL  REPRESENTATION   OF   THE   RELATION    BETWEEN 
TWO   VARIABLES 

114.  Introduction.  When  the  corresponding  changes  in 
two  related  quantities,  as  for  example  the  temperature  at 
successive  hours  of  the  day,  are  to  be  described,  it  is  useful  to 
rspresent  the  quantities  by  lines  and  points  (See  Art.  18). 
Such  a  representation  is  said  to  be  graphicaL  By  this  method, 
the  corresponding  changes  in  the  two  quantities  are  presented 
to  the  eye  in  a  very  vivid  way. 

For  example,  a  daily  paper  gives  the  following  temperatures  for  Chi- 
cago at  successive  hours  of  a  certain  November  day  : 


2  A.M. 

21° 

8  A.M. 

23° 

2  P.M. 

39° 

8  P.M. 

37° 

3  A.M. 

21° 

9  A.M. 

28° 

3  P.M. 

41° 

9  P.M. 

37° 

4  A.M. 

20° 

10  A.M. 

31° 

4  P.M. 

40° 

10  P.M. 

38° 

5  A.M. 

20° 

11  A.M. 

33° 

5  P.M. 

39° 

11  P.M. 

38° 

6  A.M. 

21° 

Noon. 

35° 

6  P.M. 

39° 

Midnight. 

37° 

7  A.M. 

22° 

1  P.M. 

38° 

7  P.M. 

38° 

1  A.M. 

36° 

The  changes  in  temperature  with  respect  to  time  are  readily  grasped  by 
the  representation  of  these  numbers  on  cross  ruled  paper  (See  Fig.  22). 

115.  Axes,  Coordinates.  In  graphical  work,  much  use  is 
made  of  two  fixed  perpendicular  lines  of  reference.  The  lines 
of  reference  X'X  and  Y'Y  (See  Fig.  23)  are  called  coordinate 
axes  and  their  intersection  is  called  the  origin. 

The  horizontal  line  X'X  is  called  the  A^-axis  and  Y'Y  is 

called  the  F-axis.      The  horizontal  distance  from   the  F-axis 

to  a  point  P  is  called  the  abscissa  or  a;-value  of  the  point.     The 

vertical  distance  from  the  X-axis  to  P  is  called  the  ordinate  or 

180 


Aht.  115] 


PLUTTIXd  OF  POINTS 


IS  I 


?/-value  of  the  point.  Tlio  .r-value  and  tlio  (/-value  of  a  point 
are  together  called  the  coordinates  of  the  point.  It  is  the 
custom  to  take  distances  measured  to  the  right  from  }''}'  as 


~ 

- 

1 

<i 

i 

,(! 

) 

h' 

\             r 

^1 

1 

- 

c 

'«)^ 

"<r 

's 

( 

- 

- 

- 

i 

> 

30° 

^ 

) 

(i 

' 

- 

(> 

) 

O0« 

^®®®^ 

10° 

- 

- 

- 

2 

A. 

M. 

7 

N 

2 

)on 

5 

P 

M. 

1 

) 

\ 

f 

Fig.  22 

r 


^^    I    I    I    I 


r 


Midnight 


^P 


O  CJ.-r.) 


182  GRAPHICAL  REPRESENTATION     [Chap.  XIX. 

positive  and  those  to  the  left  as  negative;  those  measured 
upward  from  X'X  as  positive  and  those  downward  as 
negative. 

116.  Plotting  of  points.  If  we  have  given  two  numbers, 
say  2  and  -5,  we  can  find  one  and  only  one  point  that  has  the 
first  number  for  its  abscissa  and  the  second  for  its  ordinate. 
To  find  the  location  of  the  point  for  the  numbers  x  ==  2,  y  =  -5, 
we  start  at  the  origin  0  and  measure  two  units  to  the  right 
along  the  X-axis,  and  from  this  point,  we  measure  downward 
a  distance  5.  This  point  may  be  represented  by  the  symbol 
(2,  -5).  The  symbol  (a,  h)  denotes  a  point  whose  abscissa 
is  a  and  whose  ordinate  is  b.  When  a  point  is  located  in  the 
manner  described  above,  it  is  said  to  be  plotted. 

117.  Use  of  coordinate  paper.  In  plotting  points  and 
obtaining  the  geometrical  pictures  we  are  about  to  make,  it  is 
convenient  to  use  coordinate  paper.  This  is  paper  ruled  both 
horizontally  and  vertically  as  shown  in  Figs.  22  and  24. 

EXERCISES   AND   PROBLEMS 

1.  Plot  the  points  (3,  4),  (3,  -4),  (-3,  4),  (-3,  -4). 

2.  Draw  the  triangle  whose  vertices  are  (3,  -1),  (0,  5), 
(-4,  -2). 

3.  Draw  the  quadrilateral  whose  vertices  are  (2,  -2), 
(-3,  4),  (-6,  -3),  (3,4). 


Historical  note  on  graphical  representation.  The  discovery  of  the 
method  of  representing  functions  and  equations  graphically  is  due  to 
Rene  Descartes  (159G-1650),  the  French  philosopher  and  mathematician. 
The  discovery  of  this  graphical  representation  of  equations  marks  one  of 
the  greatest  advances  ever  made  in  mathematics.  He  showed  that  dis- 
tances measured  in  opposite  directions  could  be  used  to  represent  positive 
and  negative  numbers,  and  through  such  representation  brought  mathe- 
maticians to  see  that  negative  mmibers  are  indeed  very  real  and  useful. 


Arts.  117,  118,  119]  \AHIAHLi:S  183 

4.  If  a  point  moves  parallel  to  the  A'-axis,  wliich  of  its 
coordinates  remains  constant? 

5.  If  a  point  moves  parallel  to  the  I'-axis,  wliich  of  its 
coordinates  remains  constant? 

6.  A  line  joining  two  points  is  bisected  at  the  origin.  If 
the  coordinates  of  one  end  are  (4,  5),  what  arc  the  coordinates 
of  the  other  end? 

7.  Draw  the  triangle  whose  vertices  are  (3,  0),  (0,  5), 
(-3,  -2). 

8.  Given  a  north  and  south  line,  and  an  east  and  west  line 
for  reference  lines  (  Y  and  A-axes  respectively),  the  following 
coordinates  of  points  on  a  river  indicate  its  general  cour.se  : 
(0,  -1),  (h  -2),  (1,  -2i),  (2,  -li),  (3,  1),  (4,5),  (5,  10), 
(-1,0),  (-2,1),  (-3,2),  (-31,1),  (-4,-1),  (-5,-3). 
Alap  the  river  from  x  =  -5  to  .r  =  +5. 

118.  Variables.  A  variable  is  defined  in  Art.  112.  To 
illustrate  again,  if  t  represents  the  time  of  day  measured  from 
2  A.M.  (Fig.  22),  and  T  represents  the  temperatures,  we  note 
that  each  of  these  symbols  changes  in  value  throughout  the 
24  hours.     They  are  therefore  variables. 

As  a  further  illustration,  we  may  think  of  a  particle  of 
matter  whose  position  is  given  by  (x,  y)  moving  to  various 
positions  in  the  plane  of  the  coordinate  axes.  As  such  a  parti- 
cle moves  along  a  curve  x  and  ?/  vary  in  value. 

119.  Definition  of  a  function.  If  two  variables  x  and  y 
are  so  related  that  when  a  value  of  one  is  given,  a  correspond- 
ing value  of  the  other  is  determined,  the  second  varial)le  is 
called  a  function  of  the  first.  Thus,  the  area  of  a  square  is  a 
function  of  its  side.  This  function  may  be  expressed  algebra- 
ically as 

y  =  x"", 

where  x  is  the  length  of  a  side. 


184      RELATION   BETWEEN  TWO  VARIABLES    [Chap.  XIX. 

The  simple  interest,  denoted  by  /,  earned  on  a  principal  of 
$100  at  a  rate  of  6  per  cent  per  annum  is  a  function  of  the  time 
t  in  years.     This  function  may  be  expressed  algebraically  by 

/  =  6/. 

We  have  had  many  examples  of  functions.  In  particular, 
the  idea  of  a  function  of  x  taking  different  values  when  x  changes 
has  been  well  illustrated  in  Arts.  18,  31,  and  83. 

130.  Graph  of  a  function.  By  a  method  similar  to  that 
employed  in  Ex.  8,  Art.  117,  to  map  a  river,  a  function  may 
be  represented  with  respect  to  coordinate  axes.  This  repre- 
sentation of  a  function  is  called  the  graph  of  the  function. 

Fig.  22  gives  the  graph  of  temperature  7"  as  a  function  of  the 
time  t. 

Example.  Obtain  the  graph  of  |x  +  4  for  values  of  x  between  -5 
and  +5. 

Let  ?/  =  |x  +  4.  The  object  is  to  present  a  picture  which  will  exhibit 
the  values  of  y  that  correspond  to  assigned  values  of  x.  Any  assigned 
value  of  X  with  the  corresponding  value  of  y  determines  a  point  whose 
abscissa  is  x  and  whose  ordinate  is  y. 

Assigning  values  for  x  and  computing  the  corresponding  values  for  y, 
we  obtain  the  following  table  : 

a:  I  0  I  1      I  2  I  2.5     |  3      |      4  |     5      |-1      |-1.5    |   -2  | -3      |   -4  |    -5 
J/  I  4  I  5.5  I  7  1-7.75  |  8.5  |    10  |  11.5  |    2.5  |     1.75  |       1  | -0.5  |   -2  |   -3.5 

and  so  on.     The  corresponding  values  of  x  and  y  are  plotted  as  coordi- 
nates of  points  in  Fig.  24. 

It  should  be  noted  that  there  is  no  limit  to  the  number  of 
corresponding  values  which  we  may  compute  and  imagine 
plotted  in  a  given  interval  along  the  X-axis,  and  further  that 
to  small  changes  in  the  values  of  x,  there  correspond  small 
changes  in  the  values  of  y. 

These  facts  suggest  the  idea  of  a  continuous  line  or  curve 
to  represent  the  function  much  as  a  continuous  curve  is  used 
in  mapping  a  river.  The  line  in  Fig.  24  is  the  graph  of  the 
function  |x  +  4. 


20]                             EXERC 

^ISES 

Y 

I  1          •    '    ! 

-             jl" 

-.    U-     -     4-  -^    i- 

-^^ 

^UT 

^ 

-^  _  -i-- 

i  1             / 

1             ^^ 

T                  H^ 

-l        -iv 

1    1    (P 



M   1  <ii(' 

1       A 

ij^ 

"" 

^i 

_  ^  __ 

#                 T 

;a- 

j^LT 

,t 

I 

i —          A^ 

J' 

-ii 

7 

_L 

,-'     'Ti  1 

Ml         \     / 

-^          -f- 

lLx  iLjl      _l 

Li  J  '  Jz   ■ 

jf 

1 

/ 

/ 

Two  spaces  =  1  unit 

Y' 

185 


Fig.  24 
EXERCISES 
Construct  the  graph  of  each  of  tlie  following  functions  by 
plotting  a  number  of  points  for  each  function,  and  drawing  a 
continuous  curve  through  these  points: 


1. 

3a:  +  4. 

4.    2x  -  1. 

7.   Gx  -  1. 

2. 

a:  +  G. 

5.   1  +  5. 

8.   7x+2. 

3. 

4x-3. 

6.    ir  +  |. 

186    RELATION  BETWEEN  TWO  VARIABLES    [Chap.  XIX. 

9.  The  temperature  readings  of  a  Fahrenheit  thermometer 
that  correspond  to  Centigrade  readings  are  given  by 
y  =  ^x  +  32°,  where  x  refers  to  Centigrade  readings  and  y  to 
Fahrenheit  readings.  Draw  the  graph  to  represent  Fahren- 
heit readings  to  correspond  to  given  Centigrade  readings. 

10.  A  certain  kind  of  cloth  costs  $2  per  yard.  .  What 
function  gives  the  cost  of  any  number  of  yards  if  c  is  the 
cost  and  n  the  number  of  yards?  Plot  the  graph  of  this 
function. 

11.  A  man  drives  a  car  at  the  rate  of  20  miles  per  hour. 
Write  the  function  that  gives  the  distance  d  he  drives  in  t  hours, 
and  construct  its  graph. 

12.  Butter  costs  30^  a  pound.  Draw  a  graph  to  show  the 
cost  c  of  n  pounds. 

131.  Graph  of  an  equation.  If  x  and  y  are  involved  in  an 
equation,  say 

y  -  3x  -  i  =  0,  (1) 

we  often  speak  of  the  graph  of  the  equation.  If  we  express 
y  in  terms  of  x,  we  have 

2/  =  3a:  +  4, 

and  may  plot  the  graph  of  this  function  of  x.  The  graph  of  this 
function  is  often  spoken  of  as  the  graph  of  equation  (1),  since 
coordinates  of  points  on  the  graph  and  of  no  other  points  satisfy 
the  equation. 

To  construct  the  graph  of  an  equation  in  x  and  y,  we  have 
therefore  merely  to  express  y  in  terms  of  x  and  to  construct  the 
graph  of  the  function  thus  obtained. 

The  graph  of  an  equation  is  perhaps  better  called  the  locus 
of  the  equation,  since  the  coordinates  of  points  on  the  graph 
and  of  no  other  points  satisfy  the  equation. 


AuTs.  121,122,  123]        GHAPIIK'AL  SOLUTIONS  187 

EXERCISES 

Construct  the  locus  of  each  of  tlie  followiiifjjcciuations  : 

1.  y  -  ■i.r  =  4.  4.   4a-  -  2?/  =  3.  7.   x  +  y  =  7. 

2.  y  -  X  =  a.  5.    Txr  +  4y  =  9.  B.    y  -  3j  =  2. 

3.  3x  +  iy  =  5.        6.   2x  -  5y  =  4.  9.   2y  +  Tu-  =  4. 

10.    6.r  -  3y  =  3. 

122.  Locus  of  a  linear  equation.     An  equation  of  the  form 

ax  +  by  +  c  =  0 

is  said  to  be  linear  in  x  and  y  because  its  locus  is  a  straight  line.* 
To  find  the  locus  of  a  linear  equation,  it  is  only  necessary 
to  find  two  points  of  the  locus  and  to  draw  a  straight  line 
through  them. 

Thus,  to  find  the  locus  of  a:  +  2/  =  6,  choose  j-  =  0  and  find  y  =  6; 
choose  2/  =  0,  and  find  x  =  6.  We  may  tlien  locate  the  points  (0,  G)  and 
(6,  0)  and  draw  a  straight  line  through  them. 

EXERCISES 
Draw  the  locus  of  each  of  the  following  equations  : 

1.  5x  +  6y  =  11.      3.   4a;  -  3?/  =  0.       5.   Sx  -  y  =  3. 

2.  X  +  Uy  =  1.        4.    6x  +  5y  =  4.       6.    i]y  -  x  =  8. 

7.  Write  an  equation  in  which  y  increases  as  x  increases, 
and  plot  its  locus. 

8,  Write  an  equation  in  which  y  decreases  as  x  increases, 
and  plot  its  locus. 

123.  Graphical  solution  of  equations.  If  \\v  con.struct  the 
loci  of  two  equations,  say  of  3x  -  2y  =  6  and  x  +  2?/  =  10  as 
shown  in  Fig.  25,  it  is  seen  that  the  point  of  intersection  of 
the  loci  is  (4,  3).     As  this  point  is  on  the  locus  of  each.e(iua- 


*  No  attempt  is  made  licrc  to  prove  this  statement.  The  proof  ia 
given  in  analytic  geometry.  The  fact  is  well  illustrated  in  th.-  above 
examples  and  may  be  taken  for  granted  for  the  present. 


188      RELATION  BETWEEN  TWO  VARIABLES   [Chap.  XIX. 


tion,  the  values  x  =  4,  and  y  =  3  satisfy  both  equations.  A 
pair  of  values,  such  as  (4,  3)  that  satisfies  each  of  two  equa- 
tions is  said  to  be  a  solution  of  the  pair  of  equations. 

Exercise.     Show  that  x  =  4,  y  =  3  satisfies  the  given  equations. 

Thus,  the  graphical  solution  of  two  equations  is  the  point  of 
intersection  of  the  loci  of  the  equations. 

Since  the  graph  of  a  linear  equation  is  a  straight  line,  and 
since  two  straight  lines  intersect  in  only  one  point,  there  is  in 
general  only  one  pair  of  values  of  x  and  y  that  satisfies  a  pair  of 
linear  equations  in  x  and  y. 


EXERCISES 


Plot  the  graphs  of  the  following  equations,  and  find  solu- 
tion of  each  pair  of  equations  from  the  graph.  Test  the  solution 
by  substitution  in  the  equations. 


Arts.  123,  124]      GRAPHICAL  REPRESENTATION  18U 

1.  X  +  ij  =8,  7.   Sx  +  4y  =  10, 
X  -  y  =  i.  5x  -  y  =  d. 

2.  X  -  y  =  4,  8.   5x  -  4?/  =  11, 
2x  +  5y  =  15.  4x  +  2y=U. 

3.  3.r  -  y  =  0,    '  9.    2.r  -  //  =  6, 
y  +  3.r  =  0.  4x  -2y  =  8. 

4.  0-  +  2/  +  G  =  0,  10.   2x  +  ?/  =  3, 

a:  -  ?/  =  0.  6x  -  2(/  =  14. 

5.  X  -  2?/  =  4,  11.   2x  +  ?/  =  0, 

x  +  2y  =  S.  -y  +  3x  =  0. 

6.  X  +  4?/  =  9,  12.   Sx  -  iy  =  7, 
3.r  -  ?/  =  14.  G.r  +  y  =  32. 

124.  Graphical  representation  of  scientific  data.  The 
method  of  this  chapter  for  representing  a  mathematical  function 
has  been  adopted  by  nearly  all  scientific  men  to  show  simul- 
taneous changes  in  related  quantities.  Thus,  physicists, 
chemists,  engineers,  statisticians,  economists,  historians,  and 
others,  use  graphs  to  present  to  the  eye  relations  that  could 
not  be  shown  otherwise  without  considerable  effort. 

PROBLEMS 

APPLICATIONS   IN  LIFE  INSURANCE 

1.  The  number  of  persons  per  hundred  thousand  living  at 
age  10  that  reach  certain  assigned  ages,  as  given  by  the  Ameri- 
can Experience  Table  of  Mortality,  is  shown  in  the  following 
table  to  the  nearest  thousand  : 

Ages I    10J15  I  20  |  25|30  I  35  |  40  1  45  J^50i55i60  |  65  |  70  |  75  |  80 [MJ 90 

No.  of  thousands  living|'iOO  |  96  193  |  89  1  85  |  82  |  78  |  74  |  70  |  65  |  58  |  49  |  39  1  26  |  14  |  5  |  1 

Exhil)it  the  relation  between  the  number  living  and  age. 

Suggestion:  Let  20  years  be  represented  by  1  inch  along  the  horizon- 
tal, and  20,000  persons  living  by  1  inch  along  the  vertical. 


190    RELATION  BETWEEN  TWO  VARIABLES    [Chap.  XIX. 

2.  A  man  aged  20  may  have  his  life  insured  in  a  certain 
company  by  the  payment  of  $18.50  per  year  for  SIOOO  of  in- 
surance. The  following  table  gives  the  amounts  that  must  be 
paid  if  the  insurance  is  taken  at  other  ages  : 


Age. 

Payment. 

Age. 

Payment 

25 

$20.10 

45 

$37.80 

30 

22.92 

50 

46.65 

35 

26.55 

55 

58.84 

40 

31.33 

60 

75.77 

Construct  the  graph  to  show  the  relation  between  the  payments 
and  the  age  of  the  person  when  insurance  is  taken. 

APPLICATIONS  TO  HISTORICAL  STATISTICS 

3.  The  population  of  the  United  States  as  found  from  the 
various  Censuses  is  given  by  the  following  table  : 

Year 1 1790|  1800|  1810|  1820|  1830|  1840|  1850|  1860|  1870|  1880|  1890|  1900|  1910 

Population  in  millions  1    3     |  4.3  !  7.2  |  9.6  |12.9  |17.1  |23.2  |31.4  |38.6  150.2  |62.6  |76.3  |92.0 

Represent  the  population  graphically. 

4.  The  marriage  rate  in  England  per  thousand  population 
for  years  from  1872-1890  is  given  by  the  following  : 

Year      |  1872  |  1873  |  1874  |  1875  |  1876  |  1877  |  1878  |  1879  |  1880  |  1881 
Rate       I  17.4  |  17.6  |  17.0  |  16.7  |  16.5  |  15.7   |  15.2  |  14.4  |  14.9  |  15.1 

Year      |  1882  |  1883  |  1884  |  1885  |  1886  |  1887  |  1888  |  1889  |  1890 
Rate       I  15.5  |  15.5  |  15.1   |  14.5   |  14.2  |  14.4  |  14.4   |  15.0  |  15.5 

Represent  this  table  by  a  graph. 

5.  The  following  table  gives  the  production  of  coal  in  the 
United  States  in  millions  of  tons  for  various  dates  : 

Date  I  1850  |  1855  |  1860  |  1865  |  1870  |  1875  |  1880  1  1885  |  1890  \J  895  |  mO  1 1905^|J.  9 10 

Tonsof  coall    6.3    |  11.5  |    13.  |  21.2  |  29.5  |  46.7  |  63.8  |  99.3  1 140.9"|  172.4  1240.81350.6  |447.6 

Construct  a  graph  to  show  changes  in  coal  production  from 
1850-1910. 


AuT.  124]  APPLU'ATIUNS  191 

6.  In  considering  tiie  difterent  items  that  enter  into  the 
high  cost  of  hving,  a  man  records  the  rents  per  month  he  has 
paid  for  a  house  during  the  past  twenty  years  as  follows  : 

1893    I     1896    I     1900     |     1903    |     1905     |     1909    |     1913 
S15   -l       $20    I       $24    j       $28    |       $30    |       $40    |       $55 

Construct  a  graph  to  show  increase  in  rent. 

APPLICATIONS   TO   WEATHER   REPORTS 

7.  The  following  table  gives  the  temperatures  on  a  Fahren- 
heit thermometer  at  a  certain  station  at  various  hours  of  the 
day  : 

Noon|l  P.M.I    2    I  3   |4|5|6|7|8|   9   |  10  |  11  |     12     |1a.m.[    2    [    3 
10°  I    11°   |10.5°|9.5°|8°16°|4°12°|0°|-3°|-6°|-9°|-12.5°|-15°|-18°|-20° 

4    I     5     I      6      I     7     I     8    I     9    I    10   I    11    INoonI 
-  22°|-  22°|-  21.5°|-  21°|-  20°|-  18°|-  lo°\-  12»|  -  8°| 

Construct  the  graph  to  show  the  changes  in  temperature  with 
respect  to  the  time.  Such  a  graph  is  called  a  temperature  curve. 
By  reference  to  the  graph,  give  approximately  the  lowest  and 
highest  temperatures  of  the  day.  At  about  what  time  was 
the  temperature  lowest? 

8.  Observe  the  weather  report  in  a  daily  paper,  and  draw 
the  temperature  curve  for  the  24  hours  covered  by  the  report. 


CHAPTER  XX 
SYSTEMS   OF   LINEAR  EQUATIONS 

125.  Solution  of  equations  in  two  or  more  unknowns.  By 
a  solution  of  an  equation  in  two  or  more  unknowns,  we  mean 
a  set  of  values  of  the  unknowns  that  satisfies  the  equation. 

fx  =  1 
Thus,  ^    is  a  solution  of  x  +  y  =5. 

A  single  equation  in  two  unknowns  has  many  solutions. 
Thus,  the  equation  x  +  y  =  b 

has  solutions,  I  r).(         o)>  r,l>  il.   and  so  on. 

In  fact,  an  equation  such  as  a;  +  t/  =  5  in  two  unknowns 
has  an  unlimited  number  of  solutions,  since  if  we  assign  any 
value  to  X,  we  can  find  a  corresponding  value  of  y  that  satisfies 
the  equation.  Graphically,  we  may  say  the  coordinates  of 
each  of  the  unlimited  number  of  points  on  the  line  in  Fig.  26 
satisfy  the  equation  x  +  y  =  b. 

EXERCISES 

Find  four  solutions  of  each  of  the  following  equations  : 

1.  a;  +  2^  =  9.  6.   s  +  <  =  12.  11.   2x  +  ?,y  =  5. 

2.  ?/  +  3rc  =  8.  7.   2r  +  s  =  7.  12.   8.r  -  ?/  =  6. 

3.  Ax  +  3|/  =  5.  8.   2x  -  2>y  =  10.     13.   ^x  +  Ay  =  9. 

4.  5m  +  2n  =  11.       9.   ,s'  +  5<  =  7.  14.   6.r  -  ?/  =  5. 

5.  V  -IV  ^  7.  10.    /  +  Am  =  12.       15.    /  +  m  =  10. 

192 


Arts.  126,  127,  128,  129]      SIMULTANEOUS  EC^UATIUNS       1<J3 


One  space  =  one  nnlt        Y' 

Fig.  26 

126.  Simultaneous  equations.  In  the  last  article,  we  have 
shown  that  an  equation  in  two  unknowns  has  an  unlimited 
number  of  solutions.  Two  such  equations,  say  x  +  ij  =  5  and 
2?/  —  re  =  4,  are  said  to  be  simultaneous  equations  when  at 
least  one  pair  of  values  of  x  and  y  satisfies  both. 

The  graphs  of  these  two  equations  are  shown  in  Fig.  27,  and 
we  have  found  graphically  the  solutions  of  some  simultaneous 
linear  equations  in  Art.  123.  In  the  present  chapter,  such 
equations  will  be  solved  by  algebraic  methods. 

127.  Independent  equations.  Two  equations  are  said  to 
1)0  independent  if  tlicy  have  distinct  graphs.  Thus,  the  equa- 
tions X  +y  =  5  and  2y  -  .r  =  4  (Fig.  27)  are  independent. 

128.  Dependent  or  equivalent  equations.  Two  equations 
are  dependent  or  equivalent  if  they  have  the  same  graph.  Thus, 
the  equations  x  +  y  =  ')  and  2x  +  2y  =  10  are  equivalent. 

129.  Inconsistent  equations.  Two  ('([nations  such  03 
X  +  y  =  5  and  2x  +  2y  =  IS  arc  .said  to  be  inconsistent  because 


194  SYSTEMS  OF  LINEAR  EQUATIONS     [Chap.  XX. 

Y 


\ 


X- 


^ 


:5^ 


^^' 


Y'     One  space  =  < 

Fig.  27 

there  are  no  values  of  x  and  y  that  satisfy  both  equations.  The 
graphs  oi  X  -\-  y  =  b  and  2x  +  2y  =  18  are  parallel  lines.  (Fig. 
28.) 

By  a  system  of  linear  equations,  we  mean  a  set  of  two  or 
more  linear  equations  that  are  to  be  treated  together.  A  set 
of  values  of  the  unknowns  that  satisfies  each  equation  of  a 
system  is  said  to  be  a  solution  of  the  system. 

130.  Elimination.  Combining  equations  of  a  system  in 
such  a  manner  as  to  get  rid  of  one  of  the  unknowns  is  called 
elimination. 

Example.  Find  two  numbers  such  that  2  times  the  first  plus  3  times 
the  second  equals  12,  and  4  times  the  first  minus  3  times  the  second 
equals  6. 

Solution:  Let  x  =  the  first  number, 

and  y  =  the  second  number. 

Then  2x  +  3(/  =  12,  (1) 

4x  -  3(/  =  6.  (2) 

Adding,  6x  =  18, 

x  =3. 


Arts.  130,  l;n] 


KLLMINATIOX 


195 


Substituting  3  for  x  in  (1)  gives 

6  +  3//  =  12. 
Then  3«/  =  G, 

and  2/  =  2. 

Check:  2  •  3  +  3  •  2  =  12, 

4  •  3  -  3  •  2  =  6. 
Hence  3  and  2  are  the  numbers  sought. 


\ 

^ 

^ 

\ 

\ 

?T* 

1 

V 

--K 

-!-K-! 

4-U  - 

IT    . 

i    !    ! 

1 

1  [ 

1 

-   -n"T 

:    1    1 

\    i   1    \, 

"t"  ■ 

1 

1 

j 

i-    - 

1 

1 

\       1     1 

\ 

1 

\^ 

K 

i 

1 

~1 

-^-^- 

1 

^  1  \ 

1 

s. 

\, 

s 

~^ 

\ 

\ 

\ 

^ 

n 

; 

1 

V 

1 

!      !      1 

!   1 

s 

\ 

1 

"     M 

N 

^J  1 

1 

1 

.   1  i\ 

1 

\i  1  '  "h 

1  1 

1 

_kj  1  i 

N 

v 

One  spa 

X!e=>  c 

DC  unit 

\ 

Fig.  28 

131.  Elimination  by  addition  and  subtraction.  The  exam- 
ple of  Art.  130  is  tliere  solved  i)y  a  method  of  elimination  known 
as  elimination  by  addition. 

To  apjjly  the  method  to  a  .somewhat  more  general  system, 

solve 

Sx  +  2u  =  17,  (1) 

4x  +  3/y  =  24.  (2) 

These  equations  are  marked   (1)   and   (2).     We  shall  find  it 
convenient  to  write,  for  the  sake  of  brevity,  such  statements 


196  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

as  (1)  -^  5  to  mean  the  members  of  (1)  divided  by  5,  and  (1)  •  5 
to  mean  the  members  of  (1)  multiphed  by  5,  More  generally, 
the  symbol  (n)  is  used  to  denote  the  members  of  equation 
marked  (n). 


(1)  •  3  gives                  9x  +  Qy  =  51,  (See  Art.  38) 

(3) 

(2)  •  2  gives                  Sx  +  Qy  =  48, 

(4) 

(3)  -  (4)  gives                        x  =  3. 

(5) 

Substituting  in  (1),  3  •  3  +  2y  =  17, 

(6) 

2y  =  8, 

(7) 

i/  =  4. 

(8) 

Check:  Substitute  x  =3,  y=  4  in  (1)  and  (2).     This  gives 

3  ■  3  +  2  •  4  =  17, 

4  •  3  +  3  •  4  =  24. 
Hence  a;  =  3,  y  =  4  is  the  solution  sought. 

Explanation.     In  elimination  by  addition  or  subtraction  we 

(1)  multiply  or  divide  the  members  of  the  equations  by  such 
numbers  as  will  make  the  coefficients  of  the  unknowns  to  be 
eliminated  numerically  equal. 

(2)  We  then  eliminate  by  addition  if  the  resulting  coefficients 
have  unlike  signs  and  by  subtraction  if  they  have  like  signs. 

EXERCISES 

Solve  the  following  systems  of  equations,  making  elimina- 
tions by  addition  or  sul)traction: 

5.  4.T  +  5y  =  12, 
Gx  -y  +  1Q  =  0. 

6.  3a  +  76  =  7, 
5a  +3b  =  29. 

7.  5y  -  X  =  X  +  2, 
2y  -2x  +  ^  =  0. 

8.  u  +  V  =  10, 
Zu-2v  =  10. 


1. 

-3a:  +  4y  =  10, 

X  -  y  =  1. 

2. 

5x  +  3y  =  8, 

4x  +  5y  =  -1. 

3. 

6x  -  8;/  =  20, 

5x  +  2y  =  8. 

4. 

X  +  Zy  =  5y  - 

x  +  2y  =  13. 

Arts.  131,  132]  ELIMINATION  197 

9.   ;•  +  3s  =  1,  12.    loA-  +  20/  =  10, 
4r  -  12s  =  4.  25A-  +  14/  =  11. 

10.  6/;  -  5q  =  9,  13.    Gc  +  15(/  +  G  =  0, 
7p  +  2q  =  34.  lid  +  c  =  22. 

11.  7m  -  2n  =  20,  14.    os  +  G/  =  17, 
18m  +  (jn  =  18.  6s  +  5^  =  IG. 

15.  If  the  coefficients  of  the  letter  to  be  eliminated  from 
two  simultaneous  equations  are  prime  to  each  other,  wiiat  is 
the  simplest  multipher  for  each  equation?  Answer  tiie  same 
question  if  the  coefficients  are  not  prime  to  each  other. 

16.  3x  +  4y  =  10,  18.    18u  +  lOt;  =  GO, 
5x  +  Gy  =  16.  12u  -  15y  =  105. 

17.  Ioj:  +  14y  =  18,  19.   4m  -  Ion  =  12, 
25x  -  21ij  =  163.  9m  -  10/^  =  122. 

20.    ISu  +  IOl'  =  55, 
12ii  -  15y  =  15. 

133.    Elimination  by  substitution.     This  method  of  solving 

a  system  of  linear  equations  is  iUustrated  in  the  following. 

Example:    Solve  the  system  of  equations, 

3x  -  4?/  =  14,  (1) 

5x  +  2i/  =  32.  (2) 

Solution:  From  (1),                        3x  =  4(/  +  14.  (3) 

4m  +  14 
From  (3),  x  =  -^^-^-  (4) 

Substituting    -^-- —  for  x  in  (2)  gives 

^^^^^2,=32.  (5) 

(.5)  •  3  gives                            5(4y  +  14)  +  Gy  =  96.  (6) 

Multiplying,                              20y  +70  +Gy  =  96.  (7) 

CoUecting,                                                  26//  =  26.  (8) 

2/  =  1.  (9) 
Substituting  y  =  1  in  (4)  gives 

T  =  6  (10) 

Check:  Substituting  x  =  G,  y  =  I  in  (1)  and  (2),  we  find  the  equations 
are  satisfied. 


198  SYSTEMS  OF  LINEAR  EQUATIONS   [Chap.  XX. 

Solve  the  following  systems  by  the  method  of  substitution 
and  check  the  results  by  substituting  in  the  original  system  of 
equations  : 

1.  x  +  2y  ^  11, 
5a;  -  3?/  =  3. 

2.  2x  +  y  =  8, 

3x  +  4?/  =  7. 

3.  5/i  +  2A-  =  0, 

3/i  +  /.-  =  3. 

4.  Ir  -  8s  =  5, 
r  +  lis  =  25. 


5.   8a:  +  2/  =  7, 
11a;  +  2i/  =  9. 


6.  12a;  +  by  =  14,  13.    .03a;  +  .IQy  =  10, 
3a;  -  10?/  =  26.  ■  .07.T  +  l.ly  =  18. 

1      2      5 

7.  2a; -2/=  -1,  ^*'   a;  +  ?/  "  6' 
15a;  -  5?/  =  20.  _  1  ^  1  ^  1 

a;      1/      6 

Hint:  Solve  first  for  -  and  -• 
X         y 

,,659  o  , 


7       6      2 


8. 

Ip  +  fg  =  n, 

q-p  =  2. 

9. 

3x-  10 

2        -^"^ 

^7  -  a;  =  4  . 

10. 

nh  +  5)112  =  19, 

Qnii  —  Imo  =  3. 

11. 

§.r  +  4?/  =  14, 

3a;  -  hj  =  24. 

12. 

f      59 
^^  +  3=T 

3       '^^  "  3 

5  10 


.      ^+3  =  0.  x  +  ,  =  10. 

16.   U2      7  18.1  +  1=10, 

/i      /o      6  X      y 

?      i-1.  ^  +  ^-30 


AuTs.  132,  133]      GENERAL  LINEAR   IX.il'ATlUN  199 

19.  G.r  +  3?/  =  19,  21.   4x  -  5j/  =  8, 
8.1;  -  2(/  =  L5.  Ox  +  3(/  =  12. 

20.  3.r  -  7/y  =  lo,  22.    7x  +  9/y  =  2, 
5.T  +  4y  =  11.  3x-  +  5^  =  2. 

23.   8.r  +  4y  =  49, 
7.r  -  Sy  =  17. 

133.  Standard  form  a.r  +  ft//  -f  <•  =  q.  While  some  of  the 
equations  given  in  the  following  list  of  exercises  may  appear 
complicated,  they  can  all  be  made  to  depend  on  the  solution 
of  a  standard  form, 

ax  +  by  +  c  -=  0, 

in  which  all  the  terms  containing  x,  those  containing  y,  and 
those  that  contain  neither  unknown,  are  collected.  When  the 
following  equations  are  not  given  in  this  form,  they  should 
usually  be  reduced  to  that  form  as  a  first  step  in  the  solution. 

EXERCISES 
After  reducing  the  following  equations  to  standard  form, 
solve  the  systems  by  either  process  of  elimination  : 

1.  x  -       ^      ,  °-       5      ^      3  3 
3.T  +  4y  =  15.                               X  -2y  =  1. 

2.  '^=y,  6.  ^-^-^^  =  -9, 
15x  -  9y  =  27.  lOp-q  =  2. 

..       11  7     ^-3   .?/+8_ 

'y.^        r  x  +  7   ,2y-i 


X  +  lOy  =  5.  -"2-  +  ^^y—  =  5. 

5x  +  7y  =  37 
6.r  -  2/y  =  8. 


.l+_y      x_-y  ^  8.   5x  +  7y  =  37, 

*•       2  2 


2x  -  2y  =  10. 


200  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

g    2x-  1  _  7j/-4  ^  _  13.   3p  ~2q  =  23, 

•        5  3  '  2p-l       7q  +  l 

5x  -4ij  =  12.  3^3 

a: -3  ^  ?y +  6  ^  ^  5^+  4   ,  2?/  +  1      ^ 


13       '       11 

11.  5.r  +  97/  =  40,  3a +  1      46-1       ^., 
2.r-  1      2;/-l  ^                ^^-        2       +       3       "  ^^^' 

3^2  4a  -  26  =  2. 

12.  16.r  -  4y  =  150,  .^    ^  +  1       s  -  2       , 
2x-l      ^        9.  ^^•^-^+-^  =  ^' 

-^-+^^  =  2^-  2.  +  5r  =  45. 

PROBLEMS 

1.  Find  two  numbers  whose  sum  is  100  and  whose  differ- 
ence is  28. 

2.  In  a  meeting  of  484  voters  a  motion  was  carried  by  a 
majority  of  32.     How  many  voted  aye,  and  how  many  no? 

3.  A  farmer  paid  3  men  and  2  boys  $8  for  a  day's  work, 
and  afterwards  paid  5  boys  and  2  men  $9  for  a  day's  work. 
What  were  the  wages  of  a  man  and  what  the  wages  of  a  boy? 

4.  The  admission  to  a  circus  is  50^  for  adults  and  25^  for 
children.  If  the  proceeds  from  the  sale  of  5000  tickets  is  $2200, 
how  many  tickets  of  each  kind  were  sold? 

5.  The  perimeter  of  a  rectangle  is  200  feet  and  the  length 
is  28  feet  more  than  the  width.  Find  length  and  width  of  the 
rectangle. 

6.  A  part  of  S2500  is  invested  at  6%  and  the  remainder 
at  5%.  The  yearly  income  from  both  is  $141.  Find  the 
amount  in  each  investment. 

7.  A  part  of  $5000  is  invested  at  5%  and  the  remainder  at 
4%.     The  income  from  the  part  at  4%  exceeds  that  from  the 


AiiT.  133]  PROBLEMS  201 

part  at  0%  by  $52.     Find  the  number  of  dollars  in  cadi  invest- 
ment. 

8.  If  A  should  give  B  $10,  then  B  would  have  twice  as 
much  as  ^.  If  Z^  should  give  A  $10,  then  A  and  B  woukl  have 
the  same  amounts.     How  much  money  has  each? 

9.  In  a  certain  family  each  son  has  as  many  brothers  as 
sisters,  but  each  daughter  has  twice  as  many  brothers  as  sisters. 
How  many  children  are  in  the  family? 

10.  A  bottle  and  its  contents  cost  60  cents,  and  the  contents 
cost  44  cents  more  than  the  bottle  ;  what  was  the  cost  of  the 
bottle? 

11.  A  mechanic  and  an  apprentice  together  receive  $50  for  a 
piece  of  work.  The  meclianic  works  8  days  and  the  apprentice 
12  days  ;  and  the  mechanic  earns  in  5  days  $2  more  than  the 
entire  amount  received  by  the  apprentice.  What  wages  per 
day  does  each  receive? 

12.  A  mule  and  a  donkey  were  going  to  market  laden  with 
wheat.  The  mule  said  :  "If  you  give  me  one  measure,  I  should 
carry  twice  as  much  as  you  ;  but  if  I  give  you  one,  we  should 
have  equal  burdens."     Tell  me  what  were  their  burdens. 

Tradition  says  that  Euclid  gave  this  problem  in  his  lectures  at  Alexan- 
dria 280  B.C. 

13.  A  and  B  can  do  a  piece  of  work  in  2  days  ;  but  an  equal 
piece  of  work  when  A  puts  in  only  half  his  time  and  B  only 
one-third  his  time  requires  4|  days.  How  long  would  the  work 
take  A  and  B  each,  working  alone? 

14.  A  farmer  bought  100  acres  of  land  for  .S40()().  If  part 
of  it  cost  him  $34  an  acre  and  the  remainder  $49  an  acre,  find 
the  number  of  acres  bought  at  each  jirice. 

15.  A  plumber  and  his  helper  receive  .$4.80.  The  plumber 
works  5  hours  and  the  helper  6  hours.  A\'orking  at  the  .same 
rate  per  hour  at  another  time  tiie  plumber  works  S  hours  and 
the.  helper  9^  hours,  and  they  receive  $7.()5.  What  are  the 
wages  of  each  per  hour? 


202  SYSTEMS  OF  LINEAR  EQUATIONS     [Chap.  XJ 

PROBLEMS  INVOLVING   THE  LEVER 


Fig.  29 

If  two  weights  w  and  W  balance  when  placed  on  a  bar  at  distances 
d  and  D  respectively  from  the  point  of  support  F  (called  the  fulcrum) 
then 

wd=W-D 

16.  Two  weights  balance  when  one  is  5  feet  and  the  other 
8  feet  from  the  fulcrum.  If  the  first  weight,  increased  by  25 
pounds,  be  placed  4  feet  from  the  fulcrum,  the  balance  is 
maintained.     Find  the  two  weights. 

17.  Two  children  weighing  35  and  49  pounds  just  balance 
a  seesaw  board  12  feet  long.  Where  is  the  support  of  the 
board  placed? 

18.  Two  children,  playing  on  a  seesaw  board  15  feet  long, 
just  balance  when  the  support  is  9  feet  from  one  end.  If  the 
child  on  the  long  end  of  the  board  weighs  50  pounds,  what  is 
the  weight  of  the  other  child? 

19.  Two  unknown  weights  balance  when  placed  8  and  10  <; 
feet  from  the  fulcrum  of  a  lever.     If  their  positions  are  reversed, 
5  pounds  10  ounces  must  be  added  to  the  lesser  weight  to 
restore  the  balance.     What  are  the  weights? 


PROBLEMS   ABOUT   DIGITS 

20.  In  an  intog(>r  of  two  digits  let  t  repi-esent  the  tens'  digit 
and  u  the  units'  digit.  The  number  is  then  10^  +  u.  (a)  What 
are  t  and  u  in  40?  (b)  In  85?  (c)  Write  each  number  in  the 
form  10^  +  w.  (r/)  What  is  lOt  +  u  \U  =  Q  and  u  =  3?  (e)  If 
t  =  1  and  u  =  9? 


Akt.  i;«]  rUOHLllMS  203 

21.  In  an  integer  of  three  digits  let  //,  /,  and  u  he  the  hun- 
dreds', tens',  and  units'  digits  respectively,  (a)  Write  the  num- 
bers, (h)  What  are  h,  t  and  u  in  865?  (c)  In  506?  (rf)  In  189? 
(e)  Write  each  of  these  numbers  in  the  form  lOOh  +  lOt  +  u. 
if)  What  is  lOOh  +  10t +  uii  h  =  9,  t  =  4,  u  =  3?  (g)  li  h  =  I, 
t  =  8,  u  =  0? 

22.  A  number  contains  two  digits.  The  units'  digit  is  3 
greater  than  the  tens'  digit.  The  number  equals  4  times  the 
sum  of  the  digits.     Find  the  number. 

23.  The  sum  of  the  digits  in  a  certain  two-digit  numl)er  is 
8.  If  18  be  added  to  the  number,  the  result  is  expressed  l)y 
the  digits  in  the  reverse  order.     Find  the  number. 

24.  Two  numbers  are  written  with  the  same  two  digits;  the 
difference  of  the  two  numbers  is  45  and  the  sum  of  the  digits 
is  9.     What  are  the  numbers? 

25.  The  numerator  and  denominator  of  a  certain  proper 
fraction  each  consists  of  the  same  two  figures  whose  sum  is  9, 
written  in  different  orders.  If  the  value  of  the  fraction  is  4, 
find  the  numerator  and  denominator. 

PROBLEMS   ABOUT   COINS 

26.  A  man  has  22  coins  amounting  to  SIO,  all  dollars  and 
quarters.     How  many  of  each  denomination  has  he? 

27.  A  collection  of  nickels  and  dimes,  containing  121  coins, 
amounts  to  S7.90.     How  many  coins  of  each  kind  are  there? 

28.  A  man  met  some  tramps  and  wishetl  to  give  them  a 
quarter  each,  l^ut  found  he  had  23  cents  too  little  for  that.  He 
therefore  gave  them  two  dimes  each  and  had  42  cents  over. 
How  much  money  had  he  and  how  many  tramps  were  there? 

PROBLEMS   INVOLVING    MOTION 

29.  Two  boys  run  a  race  of  440  yards.  In  the  first 
trial  A  gives  B  a  start  of  65  yards  and  wins  l)y  20  seconds.  In 
the  second  trial  A  gives  B  a  start  of  34  seconds  and  B  wins  by 
8  yards.     Find  the  rates  of  A  and  B  in  yards  per  second. 


204  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

30.  In  a  mile  race  A  gives  B  a  start  of  44  yards  and  is  beaten 
by  1  second.  In  a  second  trial  A  gives  B  a  start  of  6  seconds, 
and  beats  him  by  9|^  yards.  Find  the  number  of  yards  each 
runs  in  a  second. 

31.  A  train  traveling  30  miles  an  hour  takes  21  minutes 
longer  to  go  from  A  to  B  than  a  train  which  travels  36  miles 
an  hour.     Find  the  distance  from  A  to  B. 

32.  A  steamer  makes  50  miles  downstream  in  two  hours, 
and  returns  in  2|  hours.  Find  the  rate  of  the  current  and  the 
rate  of  the  steamer  in  still  water. 

Hint  :  Let  x  =  rate  of  steamer  in  still  water  in  miles  an  hour,  and  y 
the  rate  of  the  current.  Then  the  rate  downstream  is  x  +  y  and  the  rate 
upstream  is  x  -  y. 

33.  A  boat  goes  downstream  72  miles  in  3  hours,  and  up- 
stream 48  miles  in  3  hours.  Find  the  rate  in  still  water  and 
the  rate  of  the  current. 

PROBLEMS   ON   MIXTURES 

34.  A  grocer  has  two  kinds  of  sugar,  one  worth  5f!  and  the 
other  6^  a  pound.  How  many  pounds  of  each  sort  must  be 
taken  to  make  a  mixture  of  25  pounds  worth  $1.40? 

35.  What  quantities  of  silver  72%  pure  and  84.8%  pure 
must  be  mixed  to  give  8  ounces  of  silver  80%  pure? 

36.  If  25  pounds  of  sugar  and  10  pounds  of  coffee  together 
cost  $5.50,  and  at  the  same  price  25  pounds  of  coffee  and  10 
pounds  of  sugar  cost  $10.60,  what  is  the  price  of  each  per  pound? 

37.  A  farmer  finds  that  one  day  with  200  pounds  of  milk 
and  40  pounds  of  cream,  he  gets  24  pounds  of  butter.  On 
another  day  with  150  pounds  of  milk  and  25  pounds  of  cream, 
he  gets  16  pounds  of  butter,  (a)  What  per  cent  of  his  milk  is 
butter?     (b)  What  per  cent  of  his  cream  is  butter? 

38.  How  many  gallons  each  of  cream  40%  fat  and  milk 
5%  fat  shall  be  mixed  to  produce  30  gallons  of  the  mixture 
16|%  fat? 


Arts.  133,  134]  LITICKAL   IX^l'ATIONS  205 

39.  A  i)ouiul  of  tea  and  25  pounds  of  sugar  cost  SI.T.k  If 
sugar  rises  in  price  20  per  cent  and  tea  10  per  cent,  the  same 
amounts  cost  $2.05.     Find  tlie  price  per  pound  of  each. 

134.  Literal  equations  containing  two  unknowns.  When 
the  coefficients  of  the  unknowns  are  letters,  tiie  eciuations  with 
two  unknowns  can  still  be  solved  by  means  of  elimination,  but 
the  coefficients  of  the  unknowns  appear  in  the  results. 

EXERCISES  AND   PROBLEMS 

In  the  following  exercises,  consider  the  letters  a,  b,  c,  d 
from  the  first  of  the  alphabet  as  known  numbers  ;  solve  for 
the  X,  y,  z,  w  and  check: 

1.   x  +  3y  =  2a,  (1) 

lOx  -y  =  3a.  (2) 

Solution:   (2)  •  3  gives  30x  -  3y  =  9a.  (3) 

(1)  +  (3)  gives  31x  =  Ho,  (4) 

11a 


(5) 


Substituting  in  (1),  -^  +Sy  =  2a,  (6) 


(7) 


), 

51a 
17a 

w^^ 

17a   11a  +  51a   6 
31      31 

■"M^- 

17a   110a  -  17a 
31       31 

Check:      ^"^  +  3  • .';-  =  i±:i-J-^iir  =  ^=  2a. 

93a 

2.  ix  -2y  =  56,  4.  3.5a:  +  3y  =  5a, 
Qx  +  3y       7b.  x  +  y  =  fo. 

3.  ax  +  Sy  =  lOa,  5.  5x  -  4y  =   11a, 
4:X  -2y  =  Qa.  x  +  3y  =  Qa. 

6.  8x  +  9y  =  4a  +  9b, 

n+^y  =   7—  • 


7. 

Z         6W 

2a  ~   a 

Zz      liD      3 

a  ~  2a  ~  4' 

8. 

ax  +  by  =  0, 

X  +  y  +  c  =  0 

a.x 

+  by 

=  c 

dx  +  ey 

=  /• 

1 

1 

2o, 

X 

'^y^ 

2 

3 

36. 

X 

1/  " 

206  SYSTEMS   OF   LINEAR   EQUATIONS    [Chap.  XX. 

9. 


10. 


11.  The  base  of  a  triangle  is  10  and  the  altitude  is  8.  Find 
the  area, 

12.  The  base  of  a  triangle  is  a  and  the  altitude  is  h,  what  is 
the  area? 

13.  The  base  of  a  triangle  is  a  and  the  altitude  is  10.  What 
is  the  altitude  of  a  triangle  of  the  same  area  but  with  a  base 
a  +  2? 

14.  The  altitude  of  a  triangle  is  h  and  the  base  is  a.  If 
the  base  be  increased  by  b,  how  much  must  the  altitude  be 
decreased  so  as  to  leave  the  area  unchanged? 

15.  The  sum  of  two  numbers  is  s  ;  their  difference  is  d. 
Find  the  two  numbers. 

16.  Two  persons,  A  and  B,  can  complete  a  certain  amount 
of  work  in  8  days  ;  they  work  together  4  days  ;  B  finishes  it 
in  5  days.     Find  the  time  each  would  require  to  do  it  alone. 

17.  Two  persons,  A  and  B,  can  complete  a  certain  amount 
of  work  in  I  days  ;  they  work  together  m  days,  when  A  stops  ; 
B  finishes  it  in  n  days.  Find  the  time  each  would  require  to 
do  it  alone. 

18.  Determine  h  and  c  so  that  x~  +  hx  +  c  is  equal  to 
1  when  x  =  1  and  is  equal  to  2  when  x  =  2. 

19.  Two  books  cost  a  dollars.  The  one  cost  b  dollars  more 
than  the  other.     Find  the  cost  of  each. 

20.  If  A  gives  d  dollars  to  B,  they  have  equal  sums.'  If 
B  gives  e  dollars  to  A,  then  A  has  3  times  as  much  as  B.  How 
much  has  each? 

135.  Linear  systems  in  three  or  more  unknowns.  To  solve 
a  system  of  three  equations,  involving  three  unknowns,  one  of 


Art.  135]      EQUATIONS  IX  THREE  UXKXOWXS  207 

the  unknowns  must  be  eliminated  l)ct\veen  two  pairs  of  the  eciua- 
tions.  The  problem  is  then  reduced  to  one  of  two  unknowns. 
Likewise,  to  solve  a  system  of  four  equations,  involving  four 
unknowns,  one  unknown  must  be  selected  for  elimination, 
and  it  must  be  eliminated  from  three  pairs  of  the  equations. 
We  have  then  three  eciuations  with  three  unknowns  and  proceed 
as  above  to  reduce  them  to  two  equations  with  two  unknowns. 

Example : 

Solve               2x  +  4//  +  32  =  8,  (1) 

X  -  ay  +  2  =  -4,  (2) 

3x  -  lOy  +52  =  -3.  (3) 

Solution:  Eliminate  one  unknowTi,  say  z,  between  (1)  and  (2),  thus  : 

(2)  -3,                      3x  -  15y  +  3z  =  -12,  (4) 

(1),                            2x  +    4y  +  32  =  8,  (5) 

(4)  -  (5),  X  -  19y  =  -20.  (6) 
Now  eliminate  z  from  (2)  and  (3)  as  follows : 

(2)  -5,                     5x  -  25y  +  5z  =  -20,  (7) 

(3),                          dx  -  lOy  +  52  =  -3,  (8) 

(7)  -  (8),                          2x  -  Uiy  =  -17.  (9) 

The  equations  (6)  and  (9)  contain  only  the  unknowns  x  and  y. 

(6)  •  2,  2x  -  3S(/  =  -40,  (10) 

(9),  2x  -  iryy  =  -17,  (H) 

(10)  -(11),  -23i/=-23,  (12) 

(12)  ^  -23,  y  =  1-  (^^) 


(14) 
(15) 

(10) 
(17) 
(18) 


8ul)stituting 

1  for  y  in  (9), 

2x  - 

15  =  -17, 

Solving  (14), 

X  =  -1. 

Suljslituling 

X  =  -1,     y  =  1 

n  (1),  we  get 

-2  +  4  + 

32  =  8, 

or 

32  =  6. 

(17)  H-  3, 

2=2. 

Check: 

-2+4+6=8, 

or       8=8; 

_ 

1  -  5  +  2  =  -4, 

or        -4  = 

-4; 

-3 

-  10  +  10  =  -3, 

or        -3  = 

-3. 

1  1  1 


208  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

EXERCISES   AND   PROBLEMS 

1.  x-y+z^n,  12_]^7 
3x  +  3i/  -  22  =  60,  '  X  y  z  4' 
lOx  —  5i/  -  32  =  0.  2      3      4      5 

2.  2>x  -  by  -  2z  =  U,  x '^  y  ~  "z  ^  2' 
5x  -  8y  -  z  =  12,  7      4      8 
0^-3.^-32  =  1.  x~y  +  -z=^' 

3.  X  +  y  +z  =  6, 
2x  —  y  —  z  =  —3  Hint:    Solve  first  for 
a;  +  2y  +  32  =  14. 

4.  4a;  -  2?/  +  2  =  -6,  6.   x  +  y  ^-  z  =  a, 

a;  +  2/  +  2  =  0,  3a:  -  12?/  -  2  =  2a, 

2a;  -  3y  +  32  =  2.  2,x  +  Zy  -  z  =  a. 

I.  x-{-y  +  z  +  w  =  0,  %.   X  +  y  +  z  -\-  w  =  b, 

a;  +  22/  -  2  +  3u;  =  0,  a;  +  2?/  +  32  +  4:W  =  17, 

2x  +  y  -  2z  +  w  ==  4:.  4a;  +  3?/  +  22  +  ly  =  8, 

2x  +  4?/  -  2  +  3m;  =  5,  a;  -  2;/  +  32  -  2m;  =  3. 

9.   Find  three  numbers  such  that  the  sums  formed  by  tak- 
ing them  in  pairs  are  30,  40  and  50. 

10.  The  sum  of  three  numbers  is  76.  The  sum  of  the  first 
and  second  is  4  greater  than  the  third  number,  and  the  differ- 
ence of  the  first  and  second  is  one-third  of  the  third  number. 
Find  the  numbers. 

II.  The  perimeter  of  a  triangle  is  74.  The  sum  of  two 
sides  is  greater  by  10  than  the  third  side,  and  the  difference  of 
the  same  two  sides  is  10  less  than  the  third  side.  Find  the 
sides  of  the  triangle. 

12.  Divide  1000  into  three  parts,  such  that  the  sum  of  the 
first,  \  of  the  second,  and  yV  of  the  third  shall  be  400  ;  and  the 
sum  of  the  second,  \  of  the  first,  and  y  q-  of  the  third  shall  be  450. 

13.  Three  cities,  connected  by  straight  roads,  are  at  the 
vertices  of  a  triangle.     From  ^  to  5  by  way  of  C  is  112  miles  ; 


Art.  135]  EXERCISES  AND  PROBLEMS  209 

from  B  to  C  by  way  of  .4  is  116  miles  ;  from  C  to  .1  hy  way  of  B 
is  10-4  miles.     How  far  apart  are  the  cities? 

14.  Separate  400  into  4  parts  such  that  if  the  first  part  be 
increased  by  9,  the  second  diminished  by  9,  the  third  multiplied 
by  9,  and  the  fourth  divided  by  9,  the  results  will  all  be  eciual. 

15.  In  a  race  of  500  yards,  A  can  beat  B  by  20  yanls,  and 
C  by  30  yards.     By  how  many  yards  can  B  beat  C? 

16.  Between  two  towns  the  road  is  level  one-half  of  the  dis- 
tance, and  the  speeds  of  a  motor  car  are  9,  20  and  18  miles  p(T 
hour  up  hill,  on  the  level,  and  down  hill.  It  takes  5^  hours  to 
go  and  6f  to  return.  What  are  the  lengths  of  the  level  and 
inclined  parts  of  the  road? 

MISCELLANEOUS   EXERCISES   AND   PROBLEMS 

Solve  the  following  equations  for  x  and  rj  when  the  solution 
is  possible.  Show  by  means  of  a  graph  why  the  solution  is 
impossible  in  Exercises  4,  6,  8,  and  15. 


1. 

6x  -3y  =  15, 

6. 

5.T  -  1       Ay  -2 

2.r  +  7y  =  45. 

6        '        5 

X  +  5   ,  ?/  +  6 

2. 

x-l      (y-3) 
2               2             ' 

7. 

2      +      3      = 

a:  -  ?y  =  2, 

x-S     y  +  4. 

2.1-  -  2//  =  10. 

2            11 

8. 

2/  +  2x  =  7, 

'           3. 

2y  -9x  =  23, 

2y  +  4x  =  4. 

x  +  5y  =  -13. 

9. 

ax  +  y  =  a  +  2, 
X  +  ay  =  2a  +  1. 

4. 

X  +  y  =  6, 

10. 

ax  +  y  =  2a, 

3.T  +  3y  =  G. 

2ax  -  y  =  a. 

X     y      . 

11. 

3x  -4y  =  2a, 

5. 

3+!  =  ^' 

4x  +  3/7  =  11a. 

'1-1  =  2 

2      8' 

12. 

ax  -  by  =  0, 

bx  -  ay  =  b-  -  a 

210  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

13.  ax  +  y  =  b,  15. 

bx  +  y  =  a. 

14.    r  -  r  =  —r'  lb. 

a  +  b      a  - b      a  +  b 
x  y  I 


x  +  3y  =  2, 

3x  +  9y  =  15. 

?  +  5  =  3, 

1/    X 

is  60,  and  such  that 

one 

a  +  b       a  -  b      a  -  b 

17.  Find  two  numl>crs  whose  sum  is  60, 
of  them  exceeds  twice  the  other  by  6. 

18.  A  board  18  feet  long  is  cut  into  two  pieces  whose  lengths 
are  in  the  ratio  1  to  3.     How  long  are  the  pieces? 

19.  A  banker  changes  $5.00  into  dimes  and  nickels.  There 
are  73  coins  in  all.  How  many  dimes  and  how  many  nickels 
are  there? 

20.  There  are  two  numbers  whose  sum  is  63.  If  the  greater 
is  divided  by  the  smaller,  the  quotient  is  2  and  the  remainder 
9.     What  are  the  numbers? 

21.  A  board  a  feet  long  is  cut  into  two  pieces  whose  lengths 
are  in  the  ratio  b  to  c.  How  long  are  the  pieces  in  terms  of  a, 
b,  and  c? 

22.  A  grocer  bought  oranges,  some  at  20  cents  a  dozen  and 
some  at  18  cents  a  dozen.  He  paid  for  all  $6.70.  He  sold  them 
at  25  cents  a  dozen  and  cleared  $2.05.  How  many  oranges 
did  he  buy  at  each  price? 

23.  After  an  examination  a  teacher  decided  to  raise  each 
grade  from  x  to  y  by  the  formula  y  =  mx  +  b,  where  m  and  b 
are  to  be  determined  by  the  facts  that  a  boy  who  made  50  is 
to  receive  65  and  one  who  made  60  is  to  receive  77.  Find  m 
and  b,  also  the  new  grade  of  a  boy  who  received  75. 

24.  A  bird  flying  with  the  wind  makes  60  miles  an  hour, 
but  when  flying  against  a  wind  half  as  strong  it  makes  only 
45  miles  an  hour.  Find  rates  of  the  two  winds.  Also  the  rate 
of  the  bird  in  still  air. 

Assumption:  It  is  to  be  assumed  that  the  rate  of  the  wind  should  be 
added  to  the  x-ate  in  still  air  when  the  bird  goes  with  the  wind  and  sub- 
tracted when  it  goes  against  the  wind. 


Art.  135]  PKOBLKMS  211 

25.  The  report  from  a  pistol  travels  1080  feet  per  second 
with  the  wind,  and  1040  against  the  same  wintl.  Find  the 
rate  of  the  wind  and  the  rate  of  sound  in  still  air. 

26.  An  aeroplane  flies  with  the  wind  at  the  rate  of  80  miles 
per  hour  and  against  a  wind  twice  as  strong  at  the  rate  of  50 
miles  per  hour.     Find  its  rate  in  still  air. 

27.  A  certain  sum  of  money  is  invested  at  5  per  cent  and 
another  at  6  per  cent.  The  annual  income  from  both  invest- 
ments is  $98.  If  the  first  sum  had  been  invested  at  6  per  cent 
and  the  second  at  5  per  cent,  the  income  would  have  been  $2 
greater.     What  are  the  two  sums  of  money? 

28.  The  grocer  sold  Mrs.  Brown  3  quarts  of  strawberries 
and  2  quarts  of  cherries  for  $0.05.  He  sold  Mrs.  Jones  2  quarts 
of  strawberries  and  5  quarts  of  cherries  for  $0.80.  Find  tlie 
price  of  a  quart  of  strawberries  and  of  a  quart  of  cherries. 

29.  The  grocer  sold  Mrs.  Brown  3  quarts  of  strawberries 
and  2  quarts  of  cherries  for  a  certain  sum  of  money.  He  sold 
Mrs.  Jones  1  quart  of  strawberries  and  5  quarts  of  cherries  for 
the  same  sum  of  money.  Compare  the  price  of  a  quart  of  straw- 
berries and  the  price  of  a  quart  of  cherries. 


212  SYSTEMS  OF  LINEAR  EQUATIONS    [Chap.  XX. 

REVIEW  EXERCISES   AND   PROBLEMS 

1.  Find  a  mean  proportional  to  each  of  the  following  pairs  of  numbers  : 
-3,  -12  ;  8,  h  ;  -n-r',  irE'  ;   12\  a'  ;  5z\  20y\ 

2.  Solve  iorx:ax=b;-=b;  a  +  x  =  b  ;  x  -  a  =  b.     Tell  in  each 

a 

case  which  of  the  principles  of  Art.  38  is  used  in  solving  the  equation. 

3.  Give  an  example  of  a  linear  equation  in  two  unknowns.  What 
is  meant  by  a  system  of  linear  equations?  Give  an  example.  Define  a 
solution  of  an  equation  in  two  unknowns.  How  many  solutions  can  be 
found  for  2x  -  y  =  10?  When  are  two  linear  equations  in  two  unknowns 
said  to  be  simultaneous?  Give  an  example.  When  inconsistent?  Give 
an  example. 

4.  What  is  the  locus  of  an  equation  of  the  form  ax  +by  =  c?  How 
many  points  must  be  found  to  determine  the  locus?  Are  the  following 
pairs  of  equations  simultaneous?     Do  their  loci  intersect? 

,  .  [x  +  y  =  4:  .,.  \x  =  4  \x  =  2  .  ,\   x  =  I 

("Mx-^=2  ^^M2/=6  ^'^M:^-=3  ^^M2^=2 

5.  What  changes  of  sign  can  be  made  in  the  fraction  -r  without  chang- 
ing its  value?     Answer  the  same  question  for  -t- 

6.  Draw  a  pair  of  coordinate  axes,  XX'  and  YY',  which  intersect  at 
0  and  divide  the  paper  into  four  parts.  Show  the  part  in  which  a  point 
is  located  if  (a)  its  coordinates  are  both  positive  ;  (b)  both  negative  ;  (c) 
the  abscissa  is  positive  and  the  ordinate  negative  ;  (d)  the  abscissa  is  nega- 
tive and  the  ordinate  positive  ;  (e)  both  zero  ;  (/)  the  abscissa  is  zero  ; 
(g)  the  ordinate  is  zero. 

7.  State  an  equation  giving  the  relation  between  x  and  y  if  (a)  y 
is  twice  as  great  as  x  ;  (b)  y  is  k  times  as  great  as  x  ;  (c)  if  y  varies  as  x. 
(d)  What  does  the  laSt  relation  become  if  2/  =  16  when  x  =  8? 

8.  If  y  varies  as  the  square  of  x,  and  y  =  8  when  x  =  2,  find  y  when 
X  has  the  values  0,  .01,  .1,  h  i  1,  2,  3,  10. 

9.  Make  a  statement  concerning  the  proportionality  of  sides,  and 
one  concerning  the  proportionality  of  areas  of  similar  triangles.  Illus- 
trate by  figures  the  meaning  of  each  statement. 

10.  The  hypotenuse  of  a  right  triangle  is  5  inches  and  one  side  is  3 
inches.  Find  the  other  .side.  The  hypotenuse  of  a  similar  triangle  is  10 
inches.     Find  the  other  sides. 


Art.  135]     REVIEW  EXERCISES  AND  PROBLEMS  213 

11.  A  man  G  feet  tall  is  standing  10  feet  from  a  lamp  post  which  is  15 
feet  high.     Find  the  length  of  his  shadow. 

12.  A  tree  casts  a  shadow  CO  feet  long  when  a  post  10  feet  high  casts 
a  shadow  8  feet  long.     How  high  is  the  tree? 

13.  Show  that  in  the  proportion  a  :  6  =  c  :  d,  the  product  of  the  moans 
divided  by  either  extreme  equals  the  other  extreme  ;  and  that  the  product 
of  the  extremes  divitled  by  cither  mean  equals  the  other  mean. 

Solve  the  following  equations  : 

14.  2x  -ix  =^x  -i  -tx+2. 


16. 

7      1     23  -  X      7       1 
X  ^  3  ~     3x      "^12     4x 

16. 

ax  +  bx  =  m  +  x. 

17. 

{a-x){b-x)  =xK 

18. 

3a;  -  5      5x  -  1      x  -  4 
5x  -  5   '  7a-  -  7      X  -  1 

19. 

5x  +  3i/  +  2  =  0, 

3x  +  2(/  +  1  =  0. 

20.  2ix  =  3\y  +  4, 
2ly  =  3ix  -  47. 

21.  X  +  1  =  2(2/  +  1), 
2/  +  2  =  4(2  +  1), 
2  +3  =  Kx  +  1). 

22.  Determine  h  and  c  in  the  equation  y  =  x^  +  fcx  +  c,  if  y  =  1  when 
X  =  -  1,  and  2/  =  5  when  x  =  1. 

23.  Determine  a,  6,  and  c  in  the  equation  y  =  ax-  +  6x  +  c,  if  (/  =  1 
when  X  =  0,  2/  =  3  when  x  =  1,  and  2/  =  6  when  x  =  2. 

24.  The  volume  of  a  cylinder  varies  as  the  square  of  the  radius  when 
its  height  is  constant.  When  the  radius  is  1,  the  volume  is  15^  P'ind 
the  volume  when  the  radius  is  G. 


CHAPTER  XXI 

SQUARE  ROOT   AND   APPLICATIONS 

136.  Definition  of  a  square  root.  A  square  root  of  a  num- 
ber is  one  of  its  two  equal  factors. 

Thus,  2  is  a  square  root  of  4,  since  2-2  =  4,  and  2a  is  a  square  root  of 
4a2,  since  2a  •  2a  =  4a-. 

Since  a?  =  (—ay  =  —a  •  -a,  it  follows  that  every  square 
has  two  square  roots,  differing  only  in  sign. 

Thus,  —  2  and  +  2  are  both  square  roots  of  4. 

137.  Radical  sign.  The  radical  sign  V  is  used  to 
indicate  the  positive  square  root  of  the  number  under  it.  When 
a  negative  root  is  to  be  taken,  the  radical  sign  is  preceded  by 
the  sign  — . 

Thus,     +\/4  or  v^  nieans  2,  and  —  \/4  means  —2. 

EXERCISES 

Find  the  following  roots  by  inspection  : 

1.  V36.                          5.    V49:  9.  V8L 

2.  -V64:                      6.    Va^-  10.  -\/225: 

3.  -Va^-                       7.    VlOa  11.  -VlM. 

4.  VT2L                        8.    -V144;  12.  +V289. 

138.  Square  root  of  monomials.  The  square  root  of  a 
product  may  be  found  by  finding  the  square  root  of  each  of  its 
factors,  and  then  taking  the  product  of  these  roots.  The  type 
form  may  be  written 

Vah  =  V(i  •  Vb. 

214 


Akts.  138,  139]  SQUARE   ROOTS  215 

This  principle  is  of  much  value  in  finding  the  square  root 
of  a  monomial,  if  it  consists  of  factors  each  of  which  is  a  square. 
In  fact,  we  may  find  the  square  root  by  simply  dividing  tiie 
exponent  of  each  factor  by  2. 

Thus,    V22o  =  v/Cr25  =  \/W^  =  3  •  5  =  15, 

and  \/d^  =  \/9  \/x«  \/y»  =  SxY- 

EXERCISES 

Find  expressions  equal  to  each  of  the  following  and  free  of 
radicals  : 

1.  Viii-  8.    -V30.r^?/V.     15.       vbVy". 

2.  ^256:  9.       VSlaW.        16.       V^^aW. 

3.  V625:  10.       v^49a26'".       17.       V-cV?^ 

4.  vT225.  11.    -V^^^¥^         18.   -V25-36xV' 


8. 

-V30.r^?/«2^ 

9. 

V81a^6^ 

10. 

v'49a26»'>. 

11. 

-Vx'-f". 

12. 
13. 

VS'xY. 

5.       a/OxV-  12.       V'a^6'"a;«.        19.       V64 -Slx^*/'". 


6.    -Vl«a"6^  13.       \/3*a:y.         20.    -  v/a^-ft-'c*. 


7.    -V52-3*-x«.     14.    -V4-x^?/*.         21.       V225a:y. 

139.    Equations  solved  by  finding  square  roots.     Since  +2 
and  -2  have  the  same  square,  4,  the  equation 


is  satisfied  by  both  +2  and  -2. 
That  is, 

±>/4  =  ±2, 

are  two  solutions  of  the  equation  x'  =  4. 

EXERCISES 
Solve  the  following  equations  : 

1.  X-  =  25.  4.   X-  =  (r-. 

2.  r^  =  121.  5.  X-  =  81a*6«. 

3.  X-  -  144  =  0.  6.  x2  =  42a<62. 


216  SQUARE  ROOT  AND  APPLICATIONS  [Chap.  XXI. 

7.  x"  =  a%\  12.   x"  -  64a266  =  0. 

8.  x"  =  mcH\  13.   a;2  =  16(a  -  hy. 

9.  a;^  -  22ba%^  =  0.  14.   x^  =  4Qa^h\ 

10.  0:2  =  36a«68.  15.   a;^  =  Slc^dl 

11.  .t2  =  100(a  +  6)2.  16.   x^  =  a^pb-^'J. 

140.  Square  roots  of  trinomials.  If  a  trinomial  is  a  perfect 
square,  its  root  is  easily  extracted  by  comparison  with  the 
familiar  formula 

a-  +  2ab  +  6^  =  (a  +  b)-, 


from  which  v  "^  +  2a6  +  6"^  =  a  +  6. 


Thus,  to  find        V^x'  +  12x(/  +  iif, 
we  may  make  the  comparison  by  putting 

y/9^  =  3x  =  o, 
and  V'iy^  =  2y  =  b. 

Since  12xy  =  2ab,  we  have 

V'9x2  +  12xy  +  4^2  =  3^  +  2(/. 

EXERCISES 

Extract  the  square  root  of  the  following  by  inspection  : 

1.  4a2  +  12ab  +  9bK  4.   4:X^  +  4xy  +  y\ 

2.  16a;2  +  24x  +  9.  5.   4.r2  +  4x  +  1. 

3.  c2  -  4ac  +  4a2.  6.   9/?^^  _  6wa:  +  x-. 

141.  Process  of  finding  the  square  root.  Given  that  a  +  b 
is  a  sciuare  root  of  o?  +  2ab  +  6'^,  it  is  well  to  follow  a  certain 
process  by  which  a  +  b  may  be  obtained  from  a-  +  2ab  +  b"^. 

PROCESS 

a*  +  2a6  +  IP  \  a  +h 


Trial  divisor  =  2a 

Complete  divisor     =  2o  +  6 


2ab  +  6= 
2a6  +  62 


Art.  141]         SC^'AKE  ROOT  OF  POLYNOMIALS  217 

To  follow  the  process  indicated,  let  us  note  that  the  first 
term,  a,  of  the  root  may  be  obtained  by  taking  the  square  root 
of  a  certain  term,  a^,  of  the  given  expression. 

If  0?  is  subtracted  from  the  given  expression,  the  remainder 
is  2ah  +  ¥. 

The  second  term,  h,  of  the  root  may  be  found  by  dividing 
a  certain  term  of  the  remainder  by  2a,  which  is  twice  the  part 
of  the  root  already  found. 

On  this  account,  twice  the  root  already  found  is  called  the 
trial  divisor. 

Since  the  remainder  2ah  +  6-  =  6(2a  +  6),  the  complete 
divisor  which  multiplied  by  6,  produces  2ah  +  6^,  is  2<i  +  b. 
The  complete  divisor  is  thus  found  by  adding  the  second  term 
of  the  root  to  the  trial  divisor. 

Before  trying  to  extract  a  square  root  of  a  polynomial,  the 
terms  should  be  arranged  according  to  ascending  or  descending 
powers  of  some  letter. 

Example  1.     Extract  the  square  root  of 

by  following  the  process  just  explained. 

PROCESS 
16x2  _  24xy  +  9i/2  I  4j  -  3j/ 

16.T2 

Trial  divisor  =  8x 


-24x2/  +  9»/' 
-24xj/  +  9j/* 


Complete  divisor    =  8x  -  3// 

Example  2.     Extract  the  square  root  of 

16x2  _  24xy  +  92/2  +  16x2  -  \2yz  +  Az'.  (1) 

In  squaring  4x  -  82/  +  2z,  we  may  treat  4x  -  82/  aa  a  single  term, 

and  write  ,  .  .  ,^^ 

\  (4x  -  82/)  +  22 }  2  =  (4x  -  82/)2  +  42(4x  -  82/)  +  Az\  (2) 

Hence,  in  extracting  the  square  root  of  (1),  we  may  find  first  the  square 
root  of  the  first  three  terms  a.s  in  Example  1.  Th(>n  to  find  the  next  term, 
22,  it  is  seen  from  (2)  that  we  should  use  2(4x  -  82/)  as  a  trial  divisor. 
That  is,  twice  the  part  of  the  root  already  found  should  be  usetl  tus  a  trial 
divisor. 


-24x?/  +  9(/2 

-24xy  +  9y^ 

16X2   - 

-  V2yz  +  43^ 

16.rz  - 

-12yz  +  42= 

218  SQUARE  ROOT  AND  APPLICATIONS  [Chap.  XXI. 

PROCESS 

16x2  -  24xy  +  9i/2  +  16.C2  -  12y2  +  42^  \Ax  -  3y  +2z 
16x2 
Trial  divisor 
=  Sx 
Complete  divisor 

r  ^^.  -.  ^y 

Trial  divisor 

=  8x  -  6y 

Complete  divisor 

=  Sx  -Qy  +2z 

Explanation.  We  first  proceed  as  when  the  root  is  a  binomial,  and 
find,  for  the  first  two  terms  in  the  root,  the  expression  4x  -  3y. 

Take  twice  this  root  already  found  as  a  trial  divisor.  Hence,  the 
trial  divisor  is  8x  -  6y,  and  the  next  term  of  the  root  is  22. 

Adding  this  to  the  trial  divisor,  we  obtain  8x  -  6y  +  2z  for  a  complete 
divisor. 

Multiplying  this  by  22,  and  subtracting  the  result  as  shown,  we  have 
no  remainder. 

In  case  the  root  contains  more  than  three  terms,  the  process  indicated 
is  continued. 

EXERCISES 

Extract  the  square  roots  of  the  following  and  verify  : 

1.  25a2  +  30a  +  9.  4.   9xhf  +  12xijz  +  iz\ 

2.  i9x^  -  28a;2  +  4.  5.   9a;^  -  ^2x^  +  49x\ 

3.  9a;2  +  30xy  +  25y\  6.   25if  -  40^/  +  16. 

7.  a^  -  2a%  +  2aH^  -  2h&  +  6^  +  c". 

8.  9a2  +  2562  +  9&  -  30a6  +  18ac  -  306c. 

9.  4rcY  +  \2x^y  +  9a;2  _  302-^2  _  2{)xif  +  2hif. 

10.  1  +  2x  +  Ix"  +  6a;''  +  9x\ 

11.  9a;*  -  \2x^y  +  34.r2(/2  _  2^xy^  +  25^^. 

12.  x^  -  \^x?  +  2\x'  +  20a;  +  4. 

13.  .^-2.3  +  1-^-1  +  ^. 

14.  4a;*  -  12a;3  -  Ix"  +  24a;  +  16. 


Arts.  1-11, 142]  SQUARE  HOOTS  219 

15.  Ox-  -  30.r  +   \  -  —  +31. 

X-        X 

16.  25.r-  +  SOxif  +  9i/  -  2.5x2  -  1..5y-2  +  0M2oz\ 

.„    a\     ,     ,   -ia-x-      2ax^      x* 

17.  _+a-3,  +  -_  +  ^+^-. 

19.  X*  -  4x3  +  5x2  _  2x  +  \. 

20.  92^  -  12^3  +  4^2  _  -  +  i  4-  6. 

21     ^V^'  +  ---^  +  -- 
2/2      x2       4        X        ?/ 

22.  a''  -  Ga^  +  150^  -  20a^  +  loa^-  -  Qa  +  I. 

23.  1  +  2x  +  3x2  +  4^3  _^.  5,p4  ^  4^5  _f.  3^6  _^  2x7  _,.  ^s 

24.  16x4  -  24x3  +  25x2  -  12x  +  4. 

25.  x2''  +  2x"ij"  +  y-\ 

14?.  Square  roots  of  numbers  expressed  in  Arabic  figures. 
The  positive  square  root  of  100  is  10  ;  of  10,000  is  100  ;  of 
1,000,000  is  1000  ;  and  so  on.  Hence,  the  square  root  of  a 
number  between  1  and  100  is  l)etween  1  and  10  ;  the  square  root 
of  a  number  between  100  and  10,000  is  between  10  and  100;  the 
square  root  of  a  number  between  10,000  and  1 ,000,000  is  Ijetween 
100  and  1000;  and  so  on.  That  is  to  say,  the  integral  part  of  the 
square  root  of  a  number  of  one  or  two  figures,  contains  one 
figure;  of  a  number  of  three  or  four  figures,  contains  two  figures; 
of  a  number  of  five  or  six  figures,  contains  three  figures  ;  and 
so  on. 

Hence,  if  an  integral  number  is  separated  into  periods  of  two 
figures  each,  beginning  at  the  right,  its  squwe  root  has  as  many 
digits  as  the  number  has  periods. 


220  SQUARE   ROOT  AND  APPLICATIONS   [Chap.  XXI. 

143.  Explanation  of  process  of  finding  square  root  in  arith- 
metic. The  process  of  finding  square  roots  of  numbers  of 
arithmetic  may  now  be  explained  by  the  process  just  used  for 
polynomials. 

Example,     Find  the  square  root  of  576. 

Solution  :  Separating  into  periods  of  two  figures  each  as  indicated  in 
Art.  142,  we  have 

576. 

There  are  thus  two  figures  in  the  integral  part  of  the  root. 

To  explain  the  solution  algebraically,  we  may  use  the  formula 

{t  +  uY  =t^  +  2tu  +  u2  =  i2  +  {2t  +  u)u, 

in  which  t  is  the  number  represented  by  tens'  digit,  and  u  is  the  number 
represented  by  units'  digit. 

The  processes  in  the  columns  at  the  right  and  left  below  are  alike  if 
t  =  20  and  u  =  A. 

t^  +  2iu  +  -u2  I  t  +u  576    I  20  +  4  =  24 

fi  4  00 

Trial  divisor  =  21 


Complete  divisor 

=  2t  +u 


2tu  +  u^  2t  =  40 

2tu  +v?  2<  +  M  =  44 


176 
176 


144.  Numbers  with  more  than  two  periods.  The  method 
just  explained  applies  to  numbers  that  separate  into  more  than 
two  periods  of  two  figures  each. 

In  making  the  application,  we  consider  t  in  the  typical  form 
(^  +  2tu  +  V?  to  represent  at  each  step  the  part  of  the  root  already 
found. 

Example.     Find  the  square  root  of  44944. 

4'49'44    I  200  +  10  +  2  =  212 
4  00  00 
2t  =  400 


2t  +u  =  410 


2t  =  420 

2t  -\-u  =  422 


4944 
4100 


844 
844 


ART.S.  144,  14-,]     S(2UARE  ROOTS  OF  DECIMALS 


221 


Omitting  zeros,  we  may  condense  the  process  so  as  to  appear  ivs 
follows : 

4'49'44   I  212 
4 


4 
41 

49 
41 

42 
422 

844 

8  44 

145.  Square  roots  of  decimals.  Decimals  are  separated 
into  periods  of  two  figures  each  from  the  decimal  point  towards 
the  right;  for,  if  the  square  root  of  a  number  has  decimal  places, 
the  number  itself  has  twice  as  many  decimal  places. 

Example.     Find  the  square  root  of  5220.0625. 


52'20.'06'25 
49 


72.25 


14 
142 

3  20 

2  84 

144 
144.2 

36.06 

28.84 

144.4 
144.45 

7. 2225 
7. 2225 

When  a  number  is  not  a  perfect  square,  we  may  annex 
periods  of  zeros  and  continue  the  process  of  root  extraction. 

EXERCISES 
Find  the  square  root  of  each  of  the  following  : 

1.  3025.  9.  .599076. 

2.  508369.  10.  87.4225. 

3.  930.25.  11.  0.01 5625. 

4.  96.4324.  12.  0.00321489. 

5.  .000625.  13.  75570.01. 

6.  2985984.  14.  20880.25. 

7.  65536.  16.  1849. 

8.  13107.9601.  16.  250500.25. 


222 


SQUARE  ROOT  AND  APPLICATIONS    [Chap.  XXI. 


17.    10.9561. 


21    ^. 
^^'   576 


19.   i-  21.   ^^  23.   '''' 


18.   0.001225. 


20. 


22. 


289 


24. 


5184 
361 


U 


ir 


u 


36 
225 

256"  '^^'   324  ""■    1600 

25.  To  illustrate  square  root  by  a  diagram,  let  us  think 
of  a  square  board  that  contains  529  square  inches  (Fig. 
30),  and  try  to  find  the  number  of  inches  in  a  side.  What  is 
the  largest  square  T  whose  side  is  a 
multiple  of  10  inches  that  can  be  cut 
from  the  board?  When  T  is  found, 
what  is  the  length  of  one  of  the  rect- 
angles U?  How  many  square  inches 
of  the  529  are  left  after  T  is  removed? 
Neglecting  U'  as  small  compared  to  the 
t/'s,  how  can  the  width  of  U  be  found 
from  the  area  and  total  length  2  X  20  = 
40  of  the  2  U's?  What  is  2  x  20  called 
in  the  process  of  finding  the  square  root?  If  a  side  of  U'  be 
added  to  the  total  length  of  the  two  U's,  what  is  the  total 
length  of  the  2  U's  and  U"!  What  is  this  number  called  in  the 
process  of  finding  square  roots? 
146.  Approximate  square  root. 
Example.     Find  the  square  root  of  2,  correct  to  two  decimal 


-20 

Fig.  30 


places. 


2.'00'00'00  I  1.414 
1 


2.4 

1.00 
.96 

2.81 

.0400 
.0281 

2.824 

.011900 
.011296 

Note.  —  To  get  the  root  correct  to  two  decimal  places,  it  is  necessary 
to  find  the  figure  in  the  third  decimal  place,  and  to  note  whether  it  is 
larger  or  smaller  than  5.  In  the  present  case,  it  is  smaller  than  5,  and 
1.41  is  the  approximate  value  sought.  That  is  to  say,  we  mean  by  "cor- 
rect to  two  decimal  places,"  the  same  as  "correct  to  the  nearest  hundredth." 


Art.  146] 


APPLICATIONS 


223 


EXERCISES 

Find  square  roots  of  tlie  following,  correct  to  two  tlecimal 
places  : 


1. 

3. 

6. 

24. 

11. 

112. 

16. 

200.02 

2. 

5. 

7. 

18. 

12. 

0.02. 

17. 

0.5. 

3. 

6. 

8. 

27. 

13. 

0.002. 

18. 

!• 

4. 

8. 

9. 

2.92. 

14. 

0.005. 

19. 

i 

5. 

10. 

10. 

0.305. 

15. 

0.307. 

20.   Show  that  V^  +  0  ^  V^  +  Vd.  * 


APPLICATION   TO   PROBLEMS   FROM   MENSURATION 

Note.     Unless  otherwise  stated,  find  approximate  values  in  the  follow- 
ing problems,  correct  to  two  decimal  places. 

Using  the  letters  as  shown  in  Fig.  31,  supply  missing  values 
of  letters  in  the  following  : 


1.  a  =  6,  6  =  8, 

2.  a  =  1,  c  =  2, 


3.  a  =  20,  6  =  V'300, 

4.  a  =  23,  6  =  20, 


5.  A  boy  having  lodged  his  kite  in  the  top  of  a  tree,  finds 
that  by  letting  out  the  whole  length  of  his  line,  which  he  knows 
is  225  feet,  it  will  reach  the  ground 
180  feet  from  the  foot  of  the  tree. 
What  is  the  height  of  the  tree? 

6.  A  tree  80  feet  high  was 
broken  off  by  a  storm,  tiie  top 
striking  the  ground  40  feet  from 
the  foot  of  the  tree,  and  the  broken 
end  resting  on  the  stump.  Assum- 
ing the  ground  to  be  a  horizontal  jilane,  what  is  the  height  of 
the  part  standing? 

7.  Two  vessels  start  from  the  same  point,  one  sailing  due 
northeast  at  the  rate  of  15  miles  an  hour,  and  the  other  due 

*  The  sign  9^  is  read  "  is  not  equal  to." 


Fig.  31 


224  SQUARE   ROOT  AND  APPLICATIONS      [Chap.  XXI. 


Fig.  32 


southeast  at  the  rate  of  18  miles  an  hour.     How  far  are  they 
apart  at  the    end    of  24    hours,    supposing    the    surface    of 

the  earth  to  be  a  plane? 

8.  Find  to  the  nearest  tenth 
of  a  foot  the  sides  of  a  square 
of  area  1200  square  feet. 

9.  Find  the  diagonal  (line 
BD  of  Fig.  32)  of  a  rectangle 
of  sides  60  feet  and  30  feet. 

10.   Find  the  diagonal  of  a 
square  of  sides  50  feet. 

11.  The  diagonal  of  a  square  is  24  feet,  what  is  the  length 
of  its  sides? 

12.  The  dotted  line  in  Fig.  33  indicates  a  path  across  a 
field.  How  many  rods  are  saved  by  taking  the  path  instead 
of  following  the  road? 

13.  Find  approximately  (to  the  nearest  tenth  of  a  rod)  the 
sides  of  a  square  having  an  area  equal  to  that  of  a  rectangle 
whose  sides  are  40  rods  and  50  rods. 

14.  Find  the  sides  of  a  square  having  an  area  equal  to  that 
of  a  triangle  of  base  70  feet 

and  altitude  35  feet.  U sei 


Facts  from  Geometry  :  (1)  The 
area  of  a  circle  is  irr^,  where  tt  = 
3.1416,  and  r  is  the  radius  of  the 
circle. 

(2)  The  surface  of  a  sphere  is 
47rr2,  where  r  is  the  radius. 

(3)  The  convex  surface  of  a 
cylinder  is  27rr/i,  where  r  is  the 
radius  of  base  and  h  the  altitude. 


Fig.  33 


15.  Find  the  radius  of  a  circle  whose  area  is  (a)  50.2656 
square  feet;  (6)  1000  square  feet. 

16.  A  horse  is  to  be  tied  to  a  post  in  a  pasture  by  means 
of  a  rope  just  long  enough  so  that  he  can  graze  over  |  acre. 


I 


Art.  14G]  PROBLEMS  225 

How  many  feet  long  should  the  rope  be  if  an  allowance  of 
three  feet  is  made  for  tying? 
(One  acre  =  43,560  square  feet.) 

17.  A  given  circle  has  a  radius  of  4  feet.  What  is  the 
radius  of  a  circle  whose  area  is  twice  that  of  the  given  circle? 

18.  A  bowl  8  inches  in  diameter  in  the  form  of  a  hemi- 
sphere is  made  by  pressing  from  a  circular  piece  of  brass.  What 
is  the  diameter  of  the  piece  of  brass  required  to  make  the  bowl? 

Assumption:  In  Problems  18  and  19,  it  is  assumed  that  the  areas  are 
the  same  before  and  after  pressing. 

19.  A  shoe-blacking  box  (without  lid)  4  inches  in  diameter 
and  I  inch  deep  is  pressed  from  a  circular  piece  of  tin.  Find 
the  diameter  of  the  piece  of  tin  required. 


CHAPTER  XXII 
RADICALS 

147.  Radical.     An  indicated  root  is  called  a  radical  exjis- 
sion,  or  simply  a  radical.     Thus,  y/a  is  a  radical  expressioi 
more  briefly  a  radical.     The  radical  sign  is  used  to  indi 
other  roots  than  square  roots  by  means  of  a  figure  called  |i( 
index  of  the  root.     Thus,  in  v^27,  the  number  3  is  the  index 
shows  that  the  third  or  cube  root  is  indicated. 

In  general,  the  nth  root  of  a  number  a,  one  of  its  n  equal  fact|s 
is  written  ^a. 

148.  Rational  and  irrational  numbers.  A  rational  num 
is  an  integer  or  the  quotient  of  two  integers.  Thus,  2,  f ,  3 
are  rational  numbers. 

Exercise.     Express  3.333  as  the  quotient  of  two  integers. 

It  is  found  useful  to  extend  our  number  system  to  cont 
numbers  that  are  not  rational  numbers.     For  example,  if 
attempt  to  find  the  side  of  a  square  whose  area  is  2,  we  ml 
write  the  result  as  \/2,  but  this  is  not  a  rational*  number. 

Stated  in  another  form,  we  have  a  solution  for  the  equati 

x"  =  2 

only  when  we  extend  our  number  system  to  include  more  th 
what  we  have  defined  as  rational  numbers. 

Any  number  which  is  not  a  rational  number  is  called 
irrational  number. 

Thus,  \/2,  \/5,  y/l^,  -^7  are  irrational  numbers. 

149.  Surds.     A  surd  is  an  irrational  number  that  is  a  re 
of  a  rational  number.     Thus,  V2,  \/3,  \/5  are  surds,  but  y 

*  See  Rietz  and  Crathorne's  College  Algebra,  p.  18. 
22G 


Vhts.  150,  151]     SLMPLIFICATIUX  OF  RADICALS  227 


te 
In  symbols 


le  /«      -y/^ 


3  not  a  surd  since  Vi  =  2,  and  \/2  +  \/3  is  not  a  surd  since 
J  +  \/S  is  not  a  a  rational  number. 

Surds  that  express  square  roots  only  are  called  quadratic 
surds. 

150.  Square  root  of  a  fraction  j-  Since  a  fraction  is  squared 
by  squaring  the  numerator  and  denominator  separately,  it 
"follows  that  the  square  root  of  a  fraction  is  given  by  extracting 
l^  the  square  root  of  the  numerator  and  denominator  separately. 

^\  151.  Simplification  of  radicals;  The  form  of  a  radical 
expression  may  often  be  changed  to  advantage  without  chang- 
ing its  value. 

3  For  example,  V^  =  V?^  =  2V2; 

and  if  we  know  that  \/2  =  1.414+, 

we  have  v^  =  2(1.414+)  =  2.828+  . 

Further,  1        \/2 

V2         2' 

•  and  the  latter  is  easier  to  compute  to  a  given  number  of  decimal  places 

•  than  the  former. 

To  continue  with  ilhis(ration.s,  let  it  be  required  to  express  y^",  approx- 

5)  imately  as  a  decimal  fraction.  Three  plans  suggest  themselves,  for  we 
may  write 

/2       ■s/2                (2           , /2          /2~3        V^ 

Vs  =^'    ""'    Vs  =  V/0^6666T,    or  ^3  =  ^3^  =  -3- 


3      V3 


v/2      1.414  + 


By  the  first  plan,       ^  =  j^.U^  =  0SIG5  -  . 
By  the  .second  plan,        V0.6666+  =  0.S1G5 -. 

X.        ,         ,•     ,      ,  Ve        2.4495-       ncicr: 

By  the  thud  plan,     -^  = 3 —  =  0.8165  - . 

The  first  plan  clearly  involves  more  labor  in  c(>ni|>iitati<)n  than  either  the 
second  or  the  third  plan. 


228  RADICALS  [Chap.  XXII. 


EXERCISES 


Find,  correct  to  two  decimal  places,  the  following  square 
roots;  do  Exercises  1  and  5  by  three  plans: 

1.  VI-       4.  VI-         7.  Vf  10.  Vf- 


2.    Vi- 

5.    Vf 

8.  VA- 

11.  VA 

3.    Vi 

6.    Vf- 

3 

''     V5- 

V5 

13.  The  area  of  a  garden  plot  in  the  form  of  a  square  is  V- 
square  rods.     Find  the  length  of  a  side  to  two  decimal  places. 

14.  A  board  in  the  form  of  a  square  contains  V-  square  feet. 
Find  the  length  of  a  side  correct  to  two  decimal  places. 

153.  Meaning  of  simplification  of  a  radical.  The  expres- 
sion under  a  radical  sign  is  often  called  a  radicand.  Expressions 
involving  square  roots  are,  in  general,  said  to  be  in  simplest 
form  when  they  are  reduced  so  that  the  radicand  (1)  is  in  in- 
tegral form,  (2)  contains  no  factor  which  is  a  perfect  square, 
and  (3)  contains  no  radical  in  the  denominator. 

For  example, 

V27   =  V9^  =  3V3, 

V^^  =  Vx*xy^!J  =  xhj\/xy, 


1 

V 

/x 

^Vx 

Vx 

Vx 

Vx 

X 

Vi  = 

-VI 

^  = 

iV3. 

and 


The  right-hand  forms  of  expression  are  simplest  for  purposes 
of  computation.  This  fact  will  be  appreciated  if  we  compute 
the  values  of  the  different  expressions  to  a  certain  number  of 
decimal  places. 


Arts.  152,  153]      EXERCISES   WITH    HADK  AI.S  229 

EXERCISES 

Given  \/2  =  1.414  +  ,  \/3  =  1.732  +  ,  compute  the  following;, 
correct  to  two  places  of  decimals: 

1.  V12.  3.    v'32.  5.    VTOS  +  Vs- 

_  1  _         _ 

2.  V27.  4.    —5.  6.    \75  +  V3. 

Simplify  the  following  : 


7.    VoO.  11.    VI  -  (h)--  15-    V^OOa'lP. 


8.  V20a^b.  12.    V2  +  (|)l  16.   y//?^  -  (^ 

9.  3Vi.  13.    v/72^.  17.    )^/~ 


10.    5V|.  14.    VQSxyh'.  18.    iVi 

Put   each  of  the  following  in  a  form  without   a  number 
written  outside  the  radical  sign: 
19.   2V3. 

Solution:  In  this  case,  2V3  =  V^  ■  V^  =  \/i2. 


20.    5V3. 

24.   SxV^r— 

-?/^. 

27.  ^  ^/i/^. 

21.    1V3. 

25.    -^v^^. 

28.    Wl 

22.    V^- 

y\  X 

26.    ~v/2. 

29.   ^V7. 

23.    SqV^P- 

153.    Addition    and    subtraction    of    radicals.     When    two 
radical   expre.ssicjns,   such   as  3\/"   :in<i   oVa,   have  the  .same 
expression  for  a  radicand  or  can  be  given  the  same  radicand 
i     by  simplification,  they  are  said  to  be  similar  radicals.     Similar 


230  RADICALS  [Chap.  XXII. 

radicals  can  be  combined  into  one  term  by    additions  and 
subtractions. 

Thus,     V'S+  \/'52  =  2x72  +  4v/2 
=  (2  +_4)V2 
=  6V2. 
Again,     V4a  +  V9rt  +  \/T6a  =  2\/a  +  3  V«  +  4-^/0 
=  ^Va. 

EXERCISES 

Combine  the  following  and  simplify  as  far  as  possible  with- 
out approximating  roots  : 

1.  Vis  +  V50  +  V32. 

2.  V12  +  V27  -  V75. 

3.  V20  +  V45  -  V80. 

4.  VT28  -  V18  +  V72. 
6.  VT08  -  V147  +  \/75. 

6.  V7UU  -  v'28  +  ves. 

7.  V32a2  _  ^g^2  ^  ^18^. 

8.  V'32a  -  VSa  +  \/18a. 

9.  Va  +  V4a  -  V166  +  \/^. 


10.    V«  -  ^  +  \/4a  -  46  +  V9a  -  96. 


11.    V(«  -  h)hn  +  V9(a  -  h)hn  -  V4(a  -  6)2m. 


12.  V32^2  +  Vi28a262  +  \/lQ,2d'b\ 

13.  \/27  +  Vl08  +  -\/45  +  VSO. 

14.  VI  +  Vl2|  +  VI  +  VU- 

15.  \/5x-2  -  50a;  +  125  +  Vbx^  -  10a;  +  5. 

16.  2\/3  -  1^12  +  4\/27. 

17.  2\/|  +  i Veo  +  Vis". 

18.  Vl  +  \/48  -  V27. 


Arts.  153,  154]  QUADRATIC  SURDS  231 

19.  V4a62  +  bV25a  +  Va{a  -  36)-. 

20.  5V300a¥  -  sV2iSabK 

21.  Find  the  sum  of  the  lengths  of  diagonals  of  three  sciuares 
of  sides  10  feet,  20  feet  and  30  feet. 

154.    Multiplication  of  quadratic  surds.     To  multiply  one 
quadratic    surd   \/a   by    another    y/b,  we   use    tiie   principle, 

Art.  138,  _  

\/7i  ■  \/b  =  \/(fb. 

That  is  to  say:    The  product  of  the  square  roots  of  two  niu7i- 
bcrs  is  the  square  root  of  their  product.  ■ 

Example  1.     Find  the  product  of  -s/3  and  \/27. 
Solution:    Vs  ■  V27  =  Vs^  =  Vsl  =  9. 
Example  2.     Multiply  2\/3  +  SVS  by  VS  -  VS. 
Solution  :  2  a/3  +  3  Vs 

V5  -    V3 


2VT5  +15 

-3V15  -    6 

-Vl5+    9 

Example  3. 

Multiply  Vo  +  Va  -  6  by  Va  -  \ 

Solution  : 

\/a  +  Va  -  b 

s/a  -  Vo  -  b 

a  +  Vo*  -  ab 

-  Vo*  -  ab  -(a-b) 

Hence,  the  product  is  b. 

EXERCISES 
Perform  indicated  operations  and  simplify  results  : 

1.  V5  ■  Vl.  4.    \/20  •  V36. 

2.  VS  ■  \/l2.  6.   Vx  ■  v/x*. 

3.  V^-Vb^-  6.    V^-Vx''. 


232  RADICALS  [Chap.  XXII. 

7.  \/5  •  Vi.  9.    Vr^'  •  Vr^. 

8.  a/I  •  Vl.  10.   y/xyz"  •  -s/x^y^z^. 

11.  (3V2  -  V3)(3V'2  +  V3). 

12.  (5V3  +  -s/7)(5\/3  -  Vl). 

13.  (2V5  +  3V3)(4\/5  -  5V3). 

14.  (\/a  +  \/6)(Va  -  V^). 

15.  (3aA  +  2aA)(5aA  -  6^6). 

16.  (V2  +  \/3  +  V5)(V'2  -  V3).   ■ 

17.  (/?-|V2)(/2  +  |V2). 

18.  Find  the  value  of  x"- \i  x  =  \/5  +  \/l0! 

5  +  \/29 


2 

■1  +\/7 


19.  Find  the  value  of  x^  +  5x  -  1  when  x  ■■ 

20.  Find  the  value  of  3x~  +  2a:  -  2  when  x 

21.  Find  the  value  of  x^  +  Qx  -  4:  for  x  =  2\/3  -  2V2. 

22.  Find  the  value  of  3x-  +  3x  -  5  when  x  =  -  -\ ^r—  • 

z        z 

23.  Does  y/S  +  \/2  satisfy  the  equation  x^  -  4x  +  1  =  0? 

24.  Does  -3  ±  \/l4  satisfy  the  equation  a;^  -  6a:  -  5  =  0? 

155.  Division  of  quadratic  surds.  In  dividing  one  radical 
expression  by  another,  the  division  is  usually  indicated  by 
using  the  dividend  as  the  numerator  and  the  divisor  as  the 
denominator  of  a  fraction. 

Thus,  to  divide  vo  by  -vb,  we  write    — ^• 

We  may  also  make  use  of  the  following  principle  :  The 
quotient  of  the  square  roots  of  two  numbers  is  the  square  root  of  the 
quotient  of  the  numbers. 

That  is,  ^!  =  v/^. 


Arts.  155,  150]  RATIOXALIZATIOX  233 

EXERCISES 

Express  each  of  the  following  us  a  fraction  under  one  radical 
sign  and  reduce  to  simplest  form : 

1     ^  3     ^^  1 

i-.     — 7^'  O.     — 7:^-  O.     — 7^- 

V3  VS  V2 

2  y^^.         4.  y^.        6  ^^ 

Vs  "  V20  *  V30 

156.  Rationalization  of  denominators.  While  we  may 
thus  indicate  (Art.  155)  the  quotient  of  two  quadratic  surds 
in  two  w\ays,  it  is  frequently  necessary  to  go  further  and  present 
the  result  in  a  simpler  form  or  find  an  approximate  value  of 
the  quotient. 

Thus,  it  may  be  necessary  to  find  the  values  of  -^=>  or  -— = —  to 

V 3        V 5  -  V3 
three  places  of  decimals. 

V2 
To  find  the  approximate  value  of  — —>  three  methods  are  shown  in 
v3 

Art.  151.     It  is  a  saving  of  work  to  multiply  the  numerator  and  denom- 
inator by  the  factor  Vs  so  as  to  make  the  denominator  rational. 

\/5 

To  find  the  appro.\imate  value  of  -     ^  by  first   extracting    the 

V  5  -  V  3 
X    square  roots  of  5  and  3,  then  subtracting  the  square  root  of  3  from  that 
of  5  and  performing  the  division,  involves  three  rather  long   operations. 
The  labor  of  two  of  these  can  be  avoided  l)y  multiplying  the  numerator 
and  denominator  by  y/Fi  +  \/3.     Thus, 


V5  -  VS  '  V5  +  V3         5 
Finding  the  value  of  '■ ^  ■  '    involves  but  one  long  operation. 

The  process  just  ex])lain('d  is  called  rationalizing  the  denomi- 
nator. 


234  RADICALS  [Chap.  XXII. 

157.  Rationalizing  factor.-  The  factor  by  which  the  terms 
of  a  fraction  are  multiphed  to  make  a  denominator  rational  is 
called  a  rationalizing  factor. 

If  the  denominator  is  of  the  form  \/a,  then  clearly  y/a  is 
a  rationalizing  factor. 

If  the  denominator  is  \/a  +  y/h,  then  \/a  -  y/h  is  a  ra- 
tionaUzing  factor. 

EXERCISES    • 

Find  equivalent  expressions  with  rational  denominators: 
,     V3  ,    3a/3  +  2\/2         ^^      2V15  -  6 


\/5  a/3  -  V2 

\/8  7    1_. 

-s/20  '   \/2  -  1 

V6  ■    \/5  -  \/3 


12. 


13. 


V5 

+  2V2 

3 

V5 

+  V2 

5  + 

2\/2 

4  - 

2V2 

1 

V7 

-V2 

3 

,     \/262a;2  „  1 

4.         . —  •  9.   ^ 7:^-  14. 

\/36x  \/3  -  V2 

V3  +  V2  3  +  \/5 

0.    — p^ 7=--         10.    7=-  IS.  ,_  .- 

\/5  -  V2  3  -  \/5  3\/5  -  2V3 

Find  the  value  of  the  following  to  four  places  of  decimals, 
avoiding  as  much  lengthy  computation  as  possible  : 

16,   ^^.  19.   ^g  +  ^. 

17.  ^^^.        20. 5-+2^: 

V?  -  Vs  4  -  2V2 

3  V3  +  2\/2  a;  /- 


Art.  158]       EQUATIONS  IXVOLVIN'G  RADICALS  235 

22.  If  a  regular  decagon  is  inscribed  in  a  circle  of  radius  r, 
one  side  of  the  decagon  is  ^(V^  -  I).  Find  the  ratio  of  the  ra- 
dius to  one  side  (correct  to  three  decimal  places). 

158.  Solution  of  equations  involving  radicals.  Equations 
involving  radicals  may  sometimes  be  conveniently  solved  by 
squaring  each  member.  This  operation  is  justified  on  the 
ground  that  each  member  is  multiplied  by  the  same  number. 

Example  1.  Solve  s^x  -1=5. 

Solution:  Square  both  members  and  we  have, 

X  -  1  =  25. 
Hence,  x  =  26. 


Check:  V26  -1=5. 

Example  2.     Solve  y/'x  -  1  +  Vx  -  4  =  2.  (1) 

Solution:  Transpose  one  of  the  radicals,  say  y/x  -  1,  then 


Vx  -  4  =  2  -  Vx  -  1. 

(2) 

Square  both  members, 

X  -  4  =  4  -  4  Vx  -  1  +  X  - 

1.     (3) 

SimpUfy, 

4\/x  -1=7. 

(4) 

Square  both  members. 

16(x  -  1)  =49. 

(5) 

Hence, 

16x  =  65, 

x=4iV. 

(6) 

Check: 

V4,V.  -4=2-  V4t»j  -  1. 

or 

\=2-\\ 

Note  that  if  radicals  are  involved,  it  is  well  to  get  a  radical 
alone  on  one  side  of  the  equation  before  squaring.  It  is  spe- 
cially important  that  all  results  be  checked  by  substitution 
when  the  members  are  squared  in  the  course  of  the  .solution. 
This  is  necessary  for  the  reason  that  solutions  may  be  intro- 


236  RADICALS  [Chap.  XXII. 

duced  by  the  operation  of  squaring  both  members  which  do  not 
satisfy  the  original  equation. 

Thus,  given  j;  =  5,  (1) 

let  us  square  Both  members. 

This  gives  x"  =  25.  (2) 

Extracting  the  square  root  of  both  members, 

X  =  ±  5. 
But  X  =  -5  does  not  satisfy  equation  (1). 

EXERCISES 

Solve  and  check  results: 


1.  \/x  -  10  =  5. 

2.  V^TTO  =  \/2a:  -  6. 

3.  a;  -  5  =  Vx-  -  4a;  +  23. 

4.  ^x"  -  7x  +  4  =  Va;2  +  x  -  20. 

5.  V a;  +  1  +  Va;  -  6  =  7. 

6.  Va;  +  9  =  1  +  V:c  -  5. 

7.  V^'^  +  7  +  a;  =  9. 

8.  6  -  \/^  =  Vx  +  10. 

9.  '  V2/  +  2O  -  V2/  +  4  =  2. 
10.  VaTTs  +  3=8-  Vx. 


^^-   V  1  +  Vs  +  V6.X  =  2. 

12.  \/x  +  a  =  yjx  +  a. 

13.  V2x  -  3a  +  a/2x  =  3\/o. 

14.  \/4a  +  x  =  2  V&  +  a;  +  V^. 


Art.  158]  EXERCISES  AND  PROBLEMS  237 

Vx  +4        Vx  +  20 


15. 


16. 


V2x  +  1       V2x  +  10 

V-r  -  g  _      yj X  _ 

■\/  X  V'.r  +  a 


17.  V^  +  2  -  Vl6  +x  =  0. 

18.  ^x  +  25  =  1  +  Vx. 

19.   The  time  i  in  seconds  for  the  complete  vibration  of  a 
simple  pendulum  is  given  by 


V32.2' 


where  /  is  the  length  (in  feet)  of  the  pendulum,  and  tt  =  3.1416. 
Solve  the  equation  for  I  in  terms  of  i  and  find  the  length  of  a 
pendulum  that  makes  a  complete  vibration  in  two  seconds. 

20.  The  velocity  of  a  falling  body  starting  from  rest  is 
given  by  y  =  ■\/2g8,  where  s  is  the  distance  passed  over  and 
g  =  32.2.  Express  s  in  terms  of  v  and  g,  and  find  the  distance  s 
that  corresponds  to  a  velocity  of  128.8  feet  per  second. 


CHAPTER  XXIII 
QUADRATIC   EQUATIONS 

159.    Quadratic  equations  solved  by  factoring.     In  Art.  84, 
we  have  solved  by  factoring  some  special  equations  of  the  form 

ax"  +  6a;  +  c  =  0, 
and  have  learned  to  call  such  equations  quadratic  equations. 
Take,  for  example,  the  equation 

2a;2  +  5x  -  3  =  0. 

The  factors  of  the  left  hand  member  are  easily  found.     They  are  2x  -\ 
and  X  +  3,  and  we  may  write  our  equation  in  the  form 

(2x  -  \){x  +3)  =0. 

Any  value  of  x  that  makes  either  factor  zero  satisfies  the  equation. 
If  a;  =1,  we  have 

(2.-i  -l)a+3)  =0-Q+3)  =0. 

Again,  if  x  =  -3  we  have 

[2(-3)-l](-3  +3)  =  [2(-3)-l]-0  =0. 
Hence,  |  and  -3  are  solutions  or  roots  of  the  given  quadratic  equation. 

EXERCISES 

Solve  the  following  equations: 

1.  x^  -  Tx  +  6  =  0.  7.  2/2  +  i  -  3  =  0. 

2.  2^2  -  3x  -  5  =  0.  8.  3a;2  -  2a;  -  8  =  0. 

3.  6a;2  -  13a;  +  6  =  0.  9.  6a;2  +  35a;  -  6  =  0. 

4.  3a;2  -  bx  =  0.  10.  3x2  +  7.^  +  2  =  0. 

5.  2y'  -  ?/  -  36  =  0.  11.  8/^  +  2/  -  1  =  0. 

6.  7s2  +  10s  -  8  =  0.  12.  2a;2  -  7a;  -  15  =  0. 

238 


Arts.  159,  100.  lOl]      COMPLETIXCJ  THE  SQUAUr:  239 

13.  8/-  -  10<  -  3  =  0.  17.  2y-  -7y  +  3  =0. 

14.  X-  -  5x  =  5a:  -  25.  18.  9  -  5s^  -  12.s  =  0. 

15.  X-  -  22x  -  48  =  0.  19.  2x~  -  30  =  9x  -  x-. 

16.  .t2  -  20.r  +  51  =  0.  20.  2x-  -  x  =  3. 

160.  Completing  the  square. 
From 

we  note  that  the  last  term  (^)  is  the  square  of  half  the  co- 
efficient of  X.     Hence,  the  expression 

X-  +  bx 
lacks  only  the  term  (^j  of  being  the  square  of  x  +  ^■ 

Therefore,  if  the  square  of  half  the  coefficient  of  x  be  added  to 
an  expression  of  the  form  x-  +  bx,  the  residt  is  the  square  of  a 
binomial. 

Such  an  addition  is  usually  spoken  of  as  completing  the 
square. 

EXERCISES 

Complete  the  square  in  each  of  the  following  : 

1.  x2  +  4x.         3.   x2  +  12x.  5.   X-  +  9x. 

2.  x~  +  8x.         4.   X-  +  3x.  6.   x^  +  x. 

7.  X-  -  2ax. 

Hint:    Make  b  =  -  2a  in  the  above  discussion. 

8.  x2  -  4x.  11.    (2x)2  +  4(2x).     14.    IGx^  +  &r. 

9.  x~  -  3x.  12.   4x2  ^  g^.  15.   9J.2  _  i2x. 
10.    (2x)-'  +  2(2x).       13.    IGx^  -  2(4x).      16.   dx'  +  9x. 

161.  Equations  solved  by  completing  the  square.  K(iua- 
tions  of  the  form  ax^  +  bx  +  c  =  0 


240  QUADRATIC  EQUATIONS         [Chap.  XXIII. 

may  be  solved  by  a  process  that  involves  completing  the  square 
on  ax^  +  hx. 

Example  1.     Solve  the  equation  x-  -  6x  -  7  =  0.  (1) 

Solution:     Transpose  -7,  and  we  have 

x^  -Qx  =  7.  (2) 

Add  9  to  each  member  to  complete  square  in  left  member. 
This  gives  x^  -  6x  +  9  =  16.  (3) 

Taking  the  square  root  of  each  member, 

a;  -3  =  ±4. 
Hence,  a;  =  7  or  -1. 

Check  by  substitution  in  (1). 

Example  2.     Solve  the  equation 

2x2  +3x  +  1  =  _2x  +4.  (1) 

Solution  :  Transpose  and  divide  each  member  by  2, 

x2+fa;=|-  (2) 

Add  II  to  each  member  to  complete  square  in  left  member. 
This  gives  x^  +  |x  +  ff  =  ff-  (3) 

Taking  square  roots,  -  x  +  f  =  ±|-  (4) 

Hence,  x  =  \,  (5) 

and  also  x  =  -3  (6) 

Check  by  substitution  in  (1). 


Solve  and  check 


EXERCISES 


1.  a;2  -  3a;  -  2  =  0.  7.  x"  +  21x  +  140  =  0. 

2.  a;2  -  6x  +  4  =  0.  8.  n{n  -  1)  =  210. 

3.  s2  -  s  -  f  =  0.  9.  2/2  -  107/  =  75. 

4.  x^  -Qx  =  40.  10.  -8  =  2.1-2  +  10:c. 

5.  rc2  -  7a:  +  6  =  0.  11.  t{t  +  4)  =  7. 

6.  3a;2  -  10a;  +  3=0.  12.  (n  +  1)^  -  8(w  +  1)  =  IG. 

162.    Solution  by  Hindu  method  of  completing  the  square. 
In  case  the  coefficient  of  .t-'  in  the  equation  is  not  unity,  as  in 

2a;2  _  4a;  -  7  =  0, 
both  members  may  be  divided  by  this  coefficient  to  obtain 
an  equation  in  which  the  coefficient  of  x~  is  unity,  and  the 
equation  may  be  solved  as  shown  in  Art.  161. 


Art.  162]  HINDU  METHOD  241 

However,  the  following  method  sometimes  avoids  the 
introduction  of  fractions  until  the  last  step  of  the  work. 

Example.     Solve  the  equation 

2x=  -  4x  =  7.  (1) 

Multipy  each  member  by  8  (four  times  the  coefficient  of  x-). 
This  gives  l&x'-  -  32x  =  56.  (2) 

or  (4x)2  -  8(4j)  =  56. 

Complete  the  square,  16x*  -  32x  +  16  =  56  +  16  =  72.         (.'3) 

Extracting  square  roots,  4x  -  4  =  ±  \/72  =  ±6\/2- 

Hence,  x  =  1  ±  ^^-  (4) 

Check  by  substitution  in  (1). 

It  should  be  noted  that  the  number  16  added  to  complete 
the  square  is  the  square  of  the  coefficient  of  x  in  the  original 
equation.  This  method  of  completing  a  square  on  ax^  +  bx 
after  multiplying  by  4a  is  known  as  the  Hindu  method. 

EXERCISES 

Solve  and  verify  by  substitution  in  each  case  : 


1. 

5a:^  -  3.r  -  2  =  0. 

16. 

3s2  +  45  =  95. 

2. 

s^  +  2s  =  120. 

17. 

(2x  -  3)2  =  6x  +  1. 

3. 

a;2  +  22x  =  -120. 

18. 

m(m  +  4)  =  7. 

4. 

P  -  11/ +  28  =  0. 

19. 

x2-2cx=  1. 

5. 

27?(n  +  4)  =42. 

20. 

15x2  _  i4x  +  3  =  0. 

6. 

6^2  -  52  -  6  =  0. 

21. 

x2  +  6x  +  5  =  0. 

7. 

5x2  -  6x  =  8. 

22. 

2x2  -  7  =  4x. 

8. 

2m2  +  3m  =  27. 

23. 

s2  +  12  =  8.S. 

9. 

0.2x2  +  0.9x  =  3.5. 

24. 

3;-2  +  r  =  200. 

10. 

0.3x2  _  o.7x  =  1. 

25. 

X  +  2  '      2x 

11. 

4x2  _  19^  =  5. 

26. 

(x  +  1)2  -  8(x  +  1)  =  16 

12. 

2s2  -  55  =  42. 

27. 

3(«;2  -v)  =21^  +  51  +  4. 

13. 

5^2  _  14^  ^  _8. 

28. 

s2  =  Is  +  2. 

14. 

7x2  +  2x  =  32. 

29. 

Sm  -  10  =  wj2 

16. 

18?^  +  6y  =  4. 

30. 

ix  -  3x2  +  2  =  0. 

242  QUADRATIC   EQUATIONS         [Chap.  XXIII. 

163.  Type  form  of  a  quadratic  equation.  The  typical  form 
of  a  quadratic  equation  is 

ax^  +  bx  +  c  =  0,  (1) 

where  a,  b,  and  c  do  not  involve  x,  and  may  have  any  values 
with  the  one  exception  that  a  may  not  equal  zero. 

Since  the  result  of  multiplying  the  members  of  an  equation 
in  this  typical  form  by  any  given  number  is  an  equation  in  the 
typical  form,  the  a,  b,  and  c  can  be  selected  in  an  indefinitely 
large  number  of  ways. 

EXERCISES 

Arrange  the  following  equations  in  the  typical  form 
ax^  +  bx  +  c  =  0,  and  indicate  the  values  of  a,  b,  and  c  in  the 
resulting  equations : 


1.  4x'-5+^x  =  ~x'  +  m. 

Solution:  By  transposing  and  collecting  terms, 
^x2+|x-(5+m)  =0, 

from  which  a  =  ^f,h  =  I,  c  =  - 

(5  +  m). 

2.   x2  +  {mx  +  6)2  =  r-. 

8.    {2x  -  3)2  =  6x  +  4. 

3    x^^{x-\y_ 
^-    9  +       16       -  ^• 

^    x^  +  ?,x  +  b          X  +  \ 
^-            2           -          3     ■ 

4.   a;2  +  {ax  +  by  =  0. 

10.    15a;2  +  2  =  Ila:. 

5.   3a;2  +  5x  -  7  =  x'  -  2x. 

11.    l-2y+y-'  =  2. 

6.    (y  -  3)2  +  4y  =  16. 

12.    4^2  -1x  =  x{x  +  1). 

7.   a^x"^  +  2cx  =  -d. 

13.    {2x  -  3)2  -  6(a:  +  1)  =  8 

14.  x{x  +  4)  =  7. 

15.  4m2a;2  +  Zk'^x^ 

-  ^mx  +  2>x  -m  +  k  =  0. 

164.    Quadratic  solved  by  formula.     The  process  of  "com- 
pleting the  square"  when  applied  to  the  typical  quadratic 

aa;2  +  6.r  +  c  =  0 


Art.  164]  SOLUTION  OF  QUADRATIC  243 

leads  to  a  very  useful  formula  which  may  be  used  to  solve  all 
quadratics,  and  which  affords  the  most  convenient  method  of 
solving  many  quadratics. 

To  obtain  the  formula  that  gives  the  .solution  of 

ax-  +  bx  +  c  =  0,  (I) 

transpose  c  and  divide  by  a.     This  gives 
.^b  c 

X-  +  -X  = 

a  a 

Add  f  —  ]    to  both  members  to  make  the  left-hand  member  a 

perfect  square.     Then 

,      b  ¥  c        b^       b"^  -  4ac 

X'  +  -X  +    -r- .,  = 1-  7"^  =  Tl~> 

a         4(7-  a      4a^  4a^ 

b\-      b^  -  4ac 


or 


Extract  the  square  root,  x  + 


4a2 

b^  _  ±  Vb^^iac 
2a  ~  2a 

-b  ±  \/b^  -  Aac 


2a 
Therefore  the  roots  of  the  general  quadratic  equation, 

ax^  -\-  bx  -\-  c  =  0, 
are  

—  64-  y/b-  —  4rtc 
2a 
and 


-^/ft2  _  ^ac 


2a 

as  may  be  verified  by  substitution  in  equation  (1). 

—  h±   \/fe2  _  4f/r 
The  expression         ^ 

may  therefore  be  used  as  a  formula  for  the  solution  of  any 
quadratic  equation. 


244  QUADRATIC   EQUATIONS        [Chap.  XXIII. 

Thus,  to  solve 

2x2  -  4x  -  7  =  0, 

we  substitute  in  the  formula,  a  =  2,  6  =  -4,  c  =  -7,  and  obtain 


4±  V16  -4-2(-7) 


2  +3\/2  ^^,2  -3V2 
Hence,  2 ^^'^ o 

are  the  roots  of  the  equation. 

These  values  of  x  are  identical  with  the  solutions  given  in  (4),  Art.  162. 

EXERCISES 

Solve  the  following  equations  by  the  use  of  the  formula, 
and  verify  by  substitution: 

1.  15a;2  -  14a;  +  3  =  0.  16.  2a;2  =  9  -  2>x. 

2.  6a;2  =  19a:  -  10.  17.  x{2x  +  3)  +  1  =  0. 

3.  x2  -  2\/3x  +  2  =  0.  18.  x^  =  -4(x  -  3). 

4.  16a;2  -  34x  +  15  =  0.  19.  4(2a:  +  5)  =  x\ 

5.  {2x  +  5)2  =  45a;.  20.  4  =  a;(3a;  +  2). 

6.  2a;2  +  7  =  ?5?.  21.   4x'-  -  3x  -2  =  0. 

7.  3r2  +  r  =  200.  22.   x{2x  +  3)  +  1  =  0. 

8.  9a;2  +  3a;  =  2.  23.    1  -  3a;  =  2x~. 

«.     x^  +  3.r  +  5  a:  +  1 

9.  6a;2  +  5a;  =  -1.  24.   ^ = J-- 

10.  7x2  ^2x  =  32.  25.   '^^^  +  "^  =  0. 

. ,       7      68a:  „^    2a;  -  2      a;  -  1 

11.  8.  +  ll+-  =  ^-  26.   5-^:^=,-^- 

12.  2?/2  -  5y  -  150  =  0.  27.    2s'-  +  3s  =  27. 

13.  4a;2  -  17a;  =  -4.  28.    {2x  -  3y  -  6{x  +  1)  +  15  =  0. 

14.  4a;2  -  a;  -  3  =  0.  29.   a;^  +  (5  -  xY  =  (5  -  2a;)2. 

15.  3a;-  +  7a;  =  110.  30.   x  +  Vx^+Q  =  14. 

Hint:  See  Art.  158. 


Art.  165]  QUADRATIC    EQUATIONS  245 

165.    The  special  quadratic  uj-  +  c  =  O.     When   i»  =  0  in 
the  typical  quadratic  equation,  the  solution  is  very  simple. 
Thus,  from  ax^  +  c  =  0,  (1) 

we  have  ax-  =  -c, 

c 

x^  =  -  -> 
a 

and  x  =  ±  V 

T      a 

Equations  of  the  type  of  (1)  arc  solved  in  Art.  139. 

MISCELLANEOUS  EXERCISES   AND   PROBLEMS 
Solve  the  following  equations  : 

1.  x^  -  llx  +  30  =  0.  16.  46-  +  12.S  +  5  =  0. 

2.  Ix"-  +  2:c  =  32.  17.  Sx'-  +  2x  -  4  =  0. 

3.  .T^  -  12x  =  28.  18.  4a;2  -  28a:  +  49  =  0. 

4.  x^  -  15x  =  0.  19.  6  -  3x  -  x-  =  0. 

5.  x'  -  25  =  0.  20.  64  +  a;2  -  16x  =  0. 

6.  x'  -  4.3x  +  3.52  =  0.  21.    ^  +  ^        ^•''  +  "^ 


3a;  -  4      3x  +  22 

7.  X-  -  0.25x  =  0.15.  22.   6^^  _  35^  -6  =  0. 

8.  s2  _  12s  +  27  =  0.  23.   7/-'  +  10<  -  8  =  0. 


9. 

s'      2s 

4  -  T  -  ^^• 

24. 

20.S2  +  ll.s  -3  =  0. 

^15 

25. 

a:  +  2      a:  +  5 

x-2       a:  -  2   '  ^• 

x-7      x-5      ^' 

11. 

4x'  -  28a:  +  45  =  0. 

26. 

X         x-S       , 

a: -3          X 

12. 

x^  +  x  -  a^  -  a  =  0. 

27. 

l+x      x-\      , 
X  -  3      a-  -  2  "  '• 

13. 

6i/  +  35?/  -  6  =  0. 

28. 

0.4x-  -  0.2x  -  0.2  =  0 

14. 

15^2  +  I62  +  4  =  0. 

29. 

x2  -  2.5x  +  1.56  -  0. 

15. 

6x'^  +  5x=  -1. 

30. 

(1  -  er)x-.-  2mx  +  m- 

246  QUADRATIC   EQUATIONS         [Chap.  XXIII. 

31.  The  sum  of  two  numbers  is  30,  and  their  product  is 
176.     Find  the  numbers. 

Solution  : 
Let  X  =  one  number. 

Then  30  -  x  =  the  other. 

Since  their  product  is  176, 
and  x(.30  -  x)  =  176. 

Solving  the  quach-atic,  x  =  8,  or  22, 

and  30  -  X  =  22,  or  8. 

Hence,  the  numbers  are  8  and  22. 

32.  Divide  50  into  two  parts  whose  product  is  600. 

33.  Divide  8.4  into  two  parts  whose  product  is  17. 

34.  Find  two  consecutive  integers  whose  product  is  72. 

35.  Find  two  consecutive  integers  the  sum  of  whose  squares 
is  145. 

36.  A  square  field  contains  10  acres.  What  are  its  dimen- 
sions? 

37.  A  rectangular  field  is  two  rods  longer  than  it  is  wide 
and  it  contains  6  acres.     What  are  its  dimensions? 

38.  Some  boys  are  given  3500  square  feet  of  land  in  rec- 
tangular form  for  basket-ball  grounds.  If  the  grounds  are  20 
feet  longer  than  they  are  wide,  what  are  the  dimensions? 

39.  The  dimensions  of  a  picture  inside  the  frame  are  14  by 
18  inches.     What  is  the  width  of  the  frame  if  the  area  of  picture 

with  frame  included  is  320  square  inches? 

40.   A   piece   of  tin  in  the  form  of  a 

square  is  taken   to   make   an   open   box. 

The  box  is  made  by  cutting  out  a  3-inch 

square  from  each  corner  of  the  piece  of 

tin  and  folding  up  the  sides.     (Fig.   34). 

The  box  thus  made   contains   192  cubic 

inches.     Find  the  length  of  the  side  of  the 
Fig.  34  ..,.,,.        ° 

origmal  piece  of  tm. 

41.   A  farmer  has  a  10-acre  wheat  field  in  the  form  of  a 

square.     In  cutting  the  wheat,   he  cuts  a  strip    of   uniform 


Arts.  IGo,  166]        QUADRATIC  FUNCTIONS  247 

width  around  the  field.     Find  the  witlth  of  the  strip  when  one- 
half  of  the  wheat  is  cut. 

42.  A  stream  flows  at  the  rate  of  5  miles  an  hour  ;  a  crew 
rows  8  miles  down  the  stream  and  back  to  the  starting  point  in 
4  hours  and  40  minutes.  What  is  the  rate  of  the  crew  in  still 
water? 

43.  The  distances  through  which  a  body  falls  in  different 
periods  of  time  are  to  each  other  as  the  squares  of  those  times. 
In  how  many  seconds  will  a  body  fall  400  feet,  the  space  it 
falls  through  in  the  first  second  being  16.1  feet? 

44.  A  and  B  distribute  S1200  each  among  some  poor  people; 
A  gives  to  40  persons  more  than  B,  but  B  gives  So  more  to 
each  person  than  A  gives;  find  the  number  of  persons  helped  by 
A  and  by  B. 

45.  A  rectangle  is  15  by  20  inches.  How  much  must  be 
added  to  the  length  to  increase  the  diagonal  3  inches? 

46.  One  leg  of  a  right  triangle  is  6  feet,  and  the  other  leg 
is  one-half  the  sum  of  the  hypotenuse  and  the  given  side.  Find 
the  sides  of  the  triangle. 

47.  A  rectangular  park  56  rods  long  and  16  rods  wide  is 
surrounded  by  a  street  of  uniform  width.  This  street  contains 
4  acres.     What  is  the  width  of  the  street? 

48.  The  circumference  of  the  fore  wheel  of  a  buggy  is  3 
feet  more  than  that  of  the  hind  wheel.  If  the  fore  wheel  makes 
125  more  revolutions  than  the  hind  wheel  in  going  a  mile,  find 
the  circumference  of  each  wheel. 

49.  What  is  the  area  of  a  square  whose  diagonal  is  one  foot 
longer  than  a  side? 

50.  An  automobile  round  trip  of  250  miles  was  made  in  1 1 
hours.  On  the  return  portion  of  the  trip,  the  speed  was  4  miles 
an  hour  more  than  on  the  outgoing  portion.  Find  the  rate  each 
way. 

166.  Graphs  of  quadratic  functions.  Any  quadratic  func- 
tion,   say    Sx^  -  5x  -  2,     may    be    represented     grapliically 


248 


QUADRATIC   EQUATIONS         [Chap.  XXIII. 


(Fig.  35).     The  graph  may  throw  considerable  hght  on  the  solu- 
tion of  the  equation 

3x2  -  5a;  -  2  =  0 
formed  by  equating  the  quadratic  function  to  zero. 

In  order  to  plot  the  graph  of  3x^  -  5x  —  2,  we  first  form  a 
table  of  corresponding  values  as  follows  (compare  Arts.  18,  120) : 

To  represent  the  numbers  in  the  table  conveniently,  it  is 
best  to  use  different  scales  for  x  and  for  the  function 
3x^  -  bx  -  2,  since  for  a  range  of  values  from  -5  to  +6  on  x, 
the  function  takes  values  from  -4  to  +98.  Plotting  points  from 
our  table  of  values  and  drawing  a  smooth  curve  through  these 
points,  we  have  the  graph  in  Fig.  35. 
3^2  -  5a;  -2 


-0.5 

-i 

-2 
-3 
-4 
-5 


-2 
-4 

0 
10 
26 
48 
76 

0 

1.25 

6 
20 
40 
66 
98 


X- 


tft 


t 


I 


T 


I 


Fig.  35 


Arts    166,  167]  IMAGINARY    NUMBERS  249 

It  should  be  observed  that  the  graph  crosses  the  X-axis  at 
two  points.  The  values  of  x  that  correspond  to  these  points 
make  the  function  zero,  and  are  therefore  the  roots  of  the 
equation  formed  by  equating  the  function  to  zero.  The 
function  becomes  zero  both  when  x  =  -  -J  and  when  x  =  2. 

EXERCISES 

1.  Solve  the  ef^uation  Sx~  -  5x  -2  =  0,  and  explain  the 
meaning  of  the  roots  by  the  use  of  the  graph  (Fig.  35). 

2.  To  ask  for  the  values  of  x  that  make  3.r-  -  5.r  -  2  equal 
to  10  means  what  on  the  graph? 

3.  Form  the  equation  whose  solution  answers  the  question 
in  Exercise  2. 

Solve  this  equation. 

Plot  the  graphs  of  the  following  functions  : 

4.  x^  -Sx  +  2.  7.   .T-'  +  4x  +  3. 

5.  X-  -  7x  +  12.  8.   3.r-  +  7x  +  4. 

6.  2.t2  +  3x  +  1.  9.   x2  -  5x  +  4. 

167.  Imaginary  numbers.  Certain  quadratic  equations,  for 
example, 

.T-  +  1  =  0,  and  .r-  -  6.r  +  15  =  0 

demand  for  their  solution  an  extension*  of  our  number  system 
to  include  the  square  roots  of  negative  numl)ers.  From  the 
equation  x"^  +  1  =0,  we  have 


and  the  equation  may  be  said  to  ask  for  a  number  wiiose  s(|uare 
is  -1.  It  is  useful  to  create  such  a  number  and  it  is  cu-stomary 
to  write  it  y/^  or  i.  That  is,  i  is  to  be  thought  of  as  a 
number  whose  square  is  -  1.     The  scjuare  roots  of  negative 

*For  other  oxtonsiori.s,  soo  Arts.  1,  2,  "Jl,  148. 


250  QUADRATIC   EQUATIONS         [Chap.  XXIII. 

numbers  are  called  imaginary  numbers,  but  we  shall  see  in  the 
chapter  devoted  to  these  numbers  in  the  advanced  course  that 
this  designation  is  a  misnomer.  It  is  usually  convenient  to  re- 
duce any  imaginary  number  to  the  form  ai,  where  a  is  a  real 
number. 

EXERCISES 

Reduce  the  following  numbers  to  the  form  ai  : 

1.  v^. 


Solution:  V-4  =  V  -  1'4=  Vi  •  V  -  1  =  2  •  V  -  1  = 

2.  V^.  7.    V^.  11-    V^. 

3.  V^16.  8.    V^.  12.       /^ 

4.  V^.  9.    V^-.  13.    V-(c'  +  d''). 


5.  V-  81. 

6.  V^.  10.    \/^2c. 

Solve  the  following  equations  : 

14.  a;-^  +  1  =  0.  17.    2a;2  +  1  =  0. 

15.  x2  +  4  =  0.  18.   Sx"-  +  2  =  0. 

16.  2x2  +  6  =  0. 

168.  Graphical  meaning  of  imaginary  roots.  It  will  be 
instructive  at  this  point  to  note  an  important  property  of  the 
graphs  of  quadratic  functions  that  give  equations  with  imagi- 
nary roots  when  the  functions  are  equated  to  zero. 


Art.  168] 


IMAGINARY  liOOTS 


251 


To  illustrate,  plot  the  graphs  of 

x"-  -  Gj  +  C 


(1)  wh(>n  C  =  0, 

(2)  when  C  =  o, 

(3)  when  C  =  9, 

(4)  when  C  =  15. 


The  four  functions  plotted  differ  only  in  the  value  of  C.  They  are 
similarly  shaped  graphs;  but,  a.s  C  is  increased  from  0  to  1.'),  the  graph  is 
simply  moved  upward  on  the  paper.  From  the  graph  for  C'  =  0,  the  roota 
of  the  equation  x-  -  6x  =  0  are  seen  to  be  0  and  6.  From  the  graph  for 
C  =  5,  the  roots  of  x=  -  6a;  +  5  =  0  are  seen  to  be  1  and  5. 

We  note  that  the  graph  for  C  =  9  merely  touches  the  A'-axis  at  the 
point  X  =  3. 

The  graph  for  C  =  15  does  not  at  all  intersect  or  touch  the  A'-axis. 
This  is  the  case  with  the  graph  of  the  function  when  the  roots  of  the  equa- 
tion formed  by  equating  the  function  to  zero  arc  imaginary  numbers. 


252  QUADRATIC    EQUATIONS         [Chap  XXIII. 

To  illustrate,  solve  the  equation 


We  find 


-  6x  +  1.5  = 

0. 

6  ±  V36  - 

-60 

2 

3±  \^^ 

or  C 

(1) 


(2) 


These  solutions  are  imaginary. 

Example.  Operating  with  i  as  with  any  other  number,  and  remcmboi;- 
ing  that  i^  =  -  1,  verify  by  substitution  in  (1),  that  3  +  i  \/6  and  3~i  \/Q 
are  solutions  of  the  equation. 

Thus, 

(3  +  iV6)2  -  6(3  +  iV6)  +  15  =  9  +  6i\/6  +  6f-  -  18  -  6iV6  +  15 

=  9-6-18  +  15 

=  0. 

EXERCISES 

Plot  graphs  of  the  following  functions  and  solve  the  equa- 
tions formed  by  equating  these  functions  to  zero: 

1.   X-  +  1.  6.  x"^  -  4:X. 

7.  3x2  +  5x  -2. 

8.  3x2  ^  5^, 

9.  3x2  ^  5^  ^  4 
10.  4.1-2  +  3x  -  1. 

PROBLEMS 

1.  The  difference  of  two  numbers  is  5,  and  the  sum  of 
their  squares  is  325  ;   what  are  the  numbers? 

2.  The  altitude  of  a  triangle  is  6  inches  more  than  its 
base,  and  the  area  is  108  square  inches.  What  is  the 
altitude? 

3.  Divide  a  line  36  inches  long  into  two  parts  such  that 
the  rectangle  whose  sides  are  equal  to  the  two  parts  has  an 
area  of  315  square  inches. 


2. 

x2  -  3x  +  2. 

3. 

x2  -  4x  +  3. 

4. 

x2  -  4x  +  4. 

5. 

x2  -  4x  +  10 

Akt.  IGS]  PROBLEMS  253 

4.  The  perimeter  of  :i  rectansle  is  24  inclies,  and  its  area 
35  square  inches.     Find  the  length  and  l)readth  of  the  rectangle. 

5.  The  perimeter  of  a  rectangle  is  14  inches  and  the  area 
is  25  square  inches.  Find  the  length  and  breadth  of  the  rec- 
tangle. Explain  why  this  solution  is  imaginary  while  that  in 
Problem  4  is  real. 

6.  A  square  pond  is  surrounded  by  a  gravel  walk  with  a 
uniform  width  of  2  yards.  The  area  of  the  walk  is  equal  to  that 
of  the  pond.     Find  the  dimensions  of  the  pond. 

7.  To  get  from  one  corner  of  a  college  quadrangle  (rec- 
tangular) to  the  opposite  corner,  I  must  go  140  yards,  around 
the  sides  ;  if  I  were  allowed  to  cut  diagonally  across  the  grass 
I  should  save  40  yards.  What  are  the  dimensions  of  the 
quadrangle? 

8.  At  what  price  per  dozen  are  eggs  selling  when,  if  the 
price  were  raised  5  cents  per  dozen,  one  would  receive  twelve 
fewer  eggs  for  a  dollar? 

9.  The  sum  of  the  base  and  altitude  of  a  triangle  is  6  inches, 
and  its  area  is  10  square  inches.  Find  the  base  and  altitude. 
Explain  why  the  results  are  imaginary. 

10.  A  certain  man  has  an  income  of  S5000  more  than  his 
exemption  of  $3000  from  income  tax.  After  deducting  a  per- 
centage for  federal  income  tax  on  the  $5000,  and  then  an  ecjual 
percentage  from  the  remainder  for  a  state  income  tax,  the 
income  is  reduced  to  $7900.50.  Find  the  rate  per  cent  of  the 
income  tax. 

11.  A  man  bought  a  number  of  $100  shares  of  R.R.  stock, 
when  they  were  at  a  certain  rate  per  cent  premium,  for  .$3500; 
and  later,  when  they  were  at  the  same  rate  per  cent  di.scount, 
sold  them  all  but  5  for  $1200.  How  many  shares  did  lie  buy, 
and  how  much  did  he  pay  per  share? 

12.  The  telegraph  poles  for  a  certain  line  are  set  at  equal 
distances.  If  there  were  4  more  per  mile,  the  di.stance  between 
them  would  be  decreased  by  66  feet.  Find  the  number  of  poles 
per  mile. 


254  QUADRATIC    EQUATIONS         [Chap.  XXIII. 

13.  A  beginner  wishing  to  simplify  {x  +  5)(x  -  2)  just  leaves 
out  the  parentheses  and  writes  a;  +  5a;  -  2.  Are  there  any  values 
of  X  for  which  the  two  expressions  are  equal? 

14.  If  a  ball  is  thrown  upward  with  an  initial  speed  of  v 
feet  per  second,  it  is  known  that  after  t  seconds  its  height  will 
be  vt  -  16.W  feet.  If  such  a  ball  is  given  an  initial  speed  of 
100  feet  per  second,  after  how  many  seconds  will  it  be  at  a 
height  of  100  feet?  After  how  many  seconds  will  it  return  to 
the  starting  point? 

15.  A  balloon  is  1  mile  from  the  ground  and  is  descending 
at  the  rate  of  5  feet  per  second  when  a  sand  bag  is  dropped. 
If  the  formula  5^  +  16.1^2  gives  the  distance  that  the  bag  will 
fall  in  t  seconds,  find  the  number  of  second?  required  for  the 
bag  to  reach  the  ground. 

16.  A  farmer  is  plowing  around  a  field  160  rods  long  and  80 
rods  wide.  How  wide  a  strip  must  he  plow  around  it  to 
make  10  acres?  Draw  a  diagram  of  the  field  and  verify 
your  result. 

17.  Find  the  side  of  a  square  field  whose  area  is  equal  to  that 
of  a  rectangular  field  whose  length  exceeds  a  side  of  the  square 
by  40  rods  and  whose  width  is  20  rods  less  than  a  side  of  the 
square. 

18.  The  base  of  a  triangle  is  6  longer  than  the  altitude,  and 
the  area  is  176.     Find  the  base  and  altitude. 

19.  In  a  right  triangle,  the  hypotenuse  is  80  inches,  and 
one  leg  is  16  inches  longer  than  the  other.     Find  the  dimensions. 

20.  Divide  the  number  20  into  two  parts  whose  squares 
are  in  the  ratio  4  to  9. 

21.  An  open  box  is  made  from  a  square  piece  of  card- 
board by  cutting  square  pieces  out  of  the  corners  and  then 
folding  up  the  flaps.  Find  the  size  of  cardboard  that  is  used 
to  make  a  box  4  inches  high  to  contain  144  cubic  inches. 

22.  From  a  cardboard  a  box  twice  as  long  as  wide  and 
containing  240  cubic  inches  is  made  by  cutting  5-inch  squares 
from  the  corners.     Find  the  dimensions  of  the  cardboard. 


.  .,^„.„.„n.:......„....  .-.,■,- ■^:v>.-.— :i:ji.^-^.t../wn:vwv^« 


Arts.  168,  169]  PROBLEMS  255 

23.  A  stream  flows  at  the  rate  of  5  miles  per  hour.  A  crew 
can  row  6  miles  with  the  stream  and  the  same  distance  i)ack 
in  f  hours.  What  is  tiie  rate  of  the  boat  in  still  water?  (See 
Problem  32,  p.  204.) 

24.  A  crew  can  row  upstream  against  a  current  of  2  miles 
an  hour,  for  a  distance  of  10  miles  upstream  and  back  again 
in  2f  hours.  What  rate  should  we  expect  the  crew  to  make  in 
still  water? 

25.  The  numerator  and  denominator  of  a  given  fraction 
together  equal  80.  If  we  increase  the  numerator  by  8  and 
decrease  the  denominator  by  8,  the  resulting  fraction  is  f  as 
large  as  the  given  fraction.     What  is  the  fraction? 

169.  Historical  note  on  quadratics.  Problems  that  involve  quad- 
ratic equations  were  solved  by  Diophantus,  the  Greek  algebraist,  who 
lived  in  Alexandria  about  .300  A.D.  But  he  gave  only  one  value  of  the 
unknown.  He  did  not  recognize  the  meaning  of  a  negative  result,  although 
he  solved  some  quadratics  by  a  method  not  unlike  that  of  completing  the 
square. 

When  Diophantus  came  upon  an  equation  both  of  whose  roots  are 
negative,  he  rejected  the  equation  as  absurd  or  impossible.  When  only 
one  negative  root  occurred,  he  merely  rejected  that  root.  When  both 
roots  were  positive,  he  took  only  the  root  that  would  be  obtained  by  taking 

the  positive  sign  before  the  radical  in  the  formula  ^ '■■    In 

fact,  Diophantus  looked  upon  a  quadratic  equation  as  having  either  one 
or  no  root. 

About  five  or  six  centuries  after  Diophantus,  the  Hindus  solved  quad- 
ratic equations,  and  observed  that  they  have  two  roots.  They  did  not 
regard  an  equation  as  absurd  because  its  roots  are  not  positive,  but  merely 
rejected  the  negative  roots  on  vague  grounds  illustrated  by  the  following  : 
In  solving  the  equation 

x"  -  45x  =  2.10, 

Bhaskar  gives  x  =  50  and  x  =  -5  for  roots,  but  he  says,  "  The  second 
value  is  in  this  case  not  to  be  taken,  for  it  is  inadequate;  people  do  not 
approve  of  negative  roots." 

The  Hindus  did,  however,  observe  that  negative  numbers  may  be 
taken  to  relate  to  debts  if  positive  numbers  relate  to  assets.  It  was  not 
until  the  work  of  Descartes  (.see  p.  181)  became  known  that  the  theory 
of  the  quadratic  was  well  understood. 


CHAPTER  XXIV 
SYSTEMS   OF   EQUATIONS   INVOLVING    QUADRATICS 

170.     Introduction.     The  typical  form  of  a  quadratic  equa- 
tion in  two  unknowns  is 

ax'^  +  hxij  +  cif  +  dx  +  ey  +  f  =  0, 

where  at  least  one  of  the  coefficients  a,  b,  or  c  is  not  zero.  Ex- 
amples of  quadratic  equations  in  two  unknowns  are 

x^  -  2a:y +  3r'  -  11  =  0, 

^  +  xy  +  2x-3=0, 

xy  -  4  =  0. 

One  such  equation  is  satisfied  by  an  indefinite  number  of  pairs 
of  values  of  x  and  y.  For  example,  each  of  the  pairs  of  values 
(0,  9),  (1,  7),  (2,  6),  (4,  5)  satisfies  the  equation, 

xy  -3x  +  2y  -  18  =  0, 

as  can  easily  be  shown  by  substitution. 

EXERCISES 

Show  that  each  of  the  following  equations  in  two  variables 
is  satisfied  by  the  pairs  of  numbers  given : 

1.  xy  -3x  +  2y  -  18  =  0,  (10,    4),     (-14,    2),     (-6,    0), 
(-3,  -9). 

2.  2x'-2xy -f~-y  +  2  =  0,    (1,    1),    (1,    -4),    (0,   -2), 
(0,  1). 

3.  x^  +7/  -1=0,  (0,  1),  (0,  -1),  (f ,  ^),  (-i  -I),  iVh,  VI). 

4.  y  -x'-  +4x  -3  =0,    (0,  3),    (1,  0),    (2,   -  1),    (3,  0), 
(2  +  V'3,  2),  (2-V3,  2). 

256 


Akts.  170,  171,  172]     SIMULTANEOUS  QUADRATICS  257 

Find  at  least  four  pairs  of  values  of  x  and  y  whieii  satisfy 
each  of  the  following  equations  : 
6.   x^  +  x-y-Q  =  0. 

Solution:  Substituting  x  =  1  in  the  equation,  there  results  I  +\-  y 
-  6  =  0,  or  2/  =  -4.  Hence,  x  =  I,  y  =  -4,  satisfies  the  equation.  Again 
substituting  x  =  2,  we  find  y  =  0.  Hence  (2,  0)  satisfies  the  equation. 
In  this  way  any  number  of  pairs  of  values  satisfying  the  equation  can  be 
found. 

6.  i/2  +  2a:  -  ?/  -  2  =  0. 

7.  2x2  +  3y2  =  4. 

8.  X-  +  xy  +  y'-  +  2x  -  y  -  2  =  0. 

9.  .r^  +  2x  +  2  =  (.T  +  1)   (y  -  2). 

171.  Solution  of  simultaneous  quadratics.  Although  there 
is  an  indefinite  number  of  pairs  of  values  of  x  and  y  which 
satisfy  one  quadratic  equation  in  two  unknowns,  yet  there  are 
never  more  than  four  pairs  which  satisfy  two  different  equations. 
For  example,  the  four  pairs  of  numbens,  (3,  4),  (-3,  4),  (3,  -4), 
(-3,  -4)  satisfy  both  the  equations, 

16^2  +  27y''  -  576  =  0, 
x^  +  ?/2  -  25  =  0. 

No  other  pairs  of  numbers  can  be  found  which  will  satisfy  both 
of  these  equations.  The  general  problem  in  systems  of  simul- 
taneous equations  involving  one  or  two  quadratics  is  the  finding 
of  all  the  pairs  of  numl)ors  which  satisfy  both  equations.  This 
problem  is  in  general  quite  difficult,  but  there  are  some  types  of 
these  equations  which  can  easily  be  solved.  A  more  extended 
discussion,  together  with  the  graphical  interpretation,  will  be 
given  in  the  second  course  in  algebra. 

172.  One  equation  linear  and  one  quadratic.  A  .system  of 
two  equations  in  two  unknowns  in  wliich  one  eciuation  is  linear 
and  the  other  is  (juadratic,  can  be  .solved  by  the  method  of 
substitution.     In  this  case  there  are  in  general  two  solutit)ns. 


258     EQUATIONS  INVOLVING  QUADRATICS    [Chap.  XXIV. 

EXERCISES 
Solve  the  following  pairs  of  equations  : 
1.   2a;2  -  xy  ^  Qy, 

x  +  2y  =  7. 

Solution  :  From  the  second  equation,  we  find 

X  =7  -2y. 
Substituting  in  the  first  and  reducing,  we  have 

lOi/2  -  Q9y  +98  =0. 
Solving  this  equation  for  y  we  find 

y  =  2  or  4.9. 
Substituting  these  values  in  the  second  equation  we  find 

X  =  3,  or  -2.8. 
The  solutions  of  the  two  equations  are  then 

(3,  2),  (-2.8,  4.9). 

Check:  f  2  •  3^  -  3  ■  2  =  6  ■  2  =  12, 

I       3+2-2=7. 

r2-  (-2.8)2  -  (-2.8)(4.9)  =6-  (4.9), 
I        -2.8  +  2  •  (4.9)  =  7. 


2. 

x'  +  y'  -  25 

=  0, 

6.    xy  =  1,            ' 

x  +  y 

=  5. 

X  +  y  =  2. 

3. 

x^  +  if  -  1 

=  0, 

7.  2x2  +  2/2  =  33, 

X 

=  y- 

2x  +  y  =  9. 

4. 

x'  +  y-  =  2, 

8.    xy  =  228, 

x  +  y  =  2. 

x  +  y  =  31. 

5. 

xy  =  l, 

2x  +  y  =  3. 

9.     xy  =  85, 
x-2y  =  7. 

10. 

xy  - 
Sx- 

-X  =  0, 

-y  =  5. 

11. 

xy  +  7y  +  6x 

-  38  =  0, 

x  +  y  =  5. 

12. 

2x' 

-  ^xy  -  y'- 
5x  +  y 

=  1, 
=  3. 

Arts.  172.  173]      SIMULTANEOUS  QUADRATICS  259 

13.   X-  +  if-  =  5(.r  +  y)  +2, 

^  =  y 

2      3 

U.   2(x  +  yr--(x  +  y){x-2y)  =70, 
2{x  +  y)  -  3(x  -  2ij)   =  2. 

15.  The  sum  of  two  numbers  is  V.  Their  product  is  ^. 
What  are  the  numbers? 

16.  If  the  figures  in  a  number  of  two  digits  are  reversed,  the 
new  number  is  27  greater  than  the  given  number.  The  product 
of  the  two  numbers  is  1300.     AVhat  is  the  given  number? 

17.  The  difference  between  the  numerator  and  the  denomi- 
nator of  an  improper  fraction  is  4.  If  2  be  added  to  both 
numerator  and  denominator  the  resulting  fraction  is  less  by  /^^ 
than  the  original  fraction.     What  is  the  fraction? 

18.  The  diagonal  of  a  rectangle  is  j  yards.  The  perimeter 
is  2  yards.     What  are  the  dimensions  of  the  rectangle? 

173.  Equations  containing  x'-  and  y-  only.  The  type  form 
of  this  case  is 

ax'  +  cy-  +  f  =  0. 

If,  instead  of  x  and  y,  we  consider  x^  and  y-  as  the  unknowns, 
the  method  of  solution  is  that  for  linear  equations.  In  general 
there  will  be  four  solutions  for  a  pair  of  equations  of  this  type. 

Solve:  EXERCISES 

1.   3x2  _  j/2  =  2, 
a;2  _  2i/  =  -41. 

Solution:  Solving  for  x^  and  if  we  find  x^  =  9,  y-  =  25. 
Hence  x  =  ±  S,  y  =  ±5.     The  four  pairs  of  numbers,  (3,  .5),  (-3,  5), 
(3,  -5),    (-3,  -5)   will  be  found  upon  substitution   to  satisfy  the  two 
equations.  ,       ,,2 

3.   I  +  I  =  20, 
2.   dx"-  -  y-  =  11,  ^        '\ 

X'  +  2t  =  22.  f  +  2  ^  ^^- 


260     EQUATIONS  INVOLVING  QUADRATICS    [Chap.  XXIV. 

6.  2^2  -  2>if  =  6, 
Sx-^  -  2i/2  =  19. 

7.  ax"-  -  hif  =  a-  +  h\ 
hx'-  +  ay-  =  a^  +  h^ 


4.   3x2  _  52^2  ^  3^ 

x'  +  y'  =  25. 

5.   x'  +  y'  =  16, 

4^2  +  25if  =  100. 

8.   nx2- 

m 

=  2 

m 

ny  = 

/I- 

9.  The  square  of  the  diagonal  of  a  rectangle  is  34.  If 
two  such  rectangles  were  put  end  to  end,  the  square  of  the 
diagonal  of  the  new  rectangle  thus  formed  is  109.  What  are 
the  dimensions  of  the  first  rectangle? 

10,  Two  rectangles  of  the  same  size  and  having  diagonals 
2|  feet  long,  are  placed  end  to  end.  The  square  on  the  diag- 
onal of  the  new  rectangle  thus  formed  is  b\  square  feet  greater 
than  the  square  on  the  diagonal  of  the  rectangle  formed  by 
putting  one  of  the  rectangles  above  the  other.  What  are  the 
dimensions  of  the  first  rectangles? 

174.  Special  methods.  The  solution  of  two  quadratic 
equations  in  two  unknowns  is  usually  in  the  nature  of  a  puzzle 
for  which  no  special  rules  are  given.  Often  no  solution  can 
be  found  without  the  use  of  mathematics  beyond  the  courses 
given  in  high  schools.  Many  systems  may  be  solved  by 
special  devices  in  which  the  aim  is  to  find  values  for  any  two  of 
the  expressions  x  +  y,  x  —  y,  and  xy,  from  which  the  values  of 
X  and  y  may  be  obtained.  Various  manipulations  are  performed 
in  attaining  this  object,  according  to  the  form  of  the  given 
equations. 

Whatever  method  is  used  in  solving  simultaneous  equations, 
it  must  be  kept  in  mind  that  the  ultimate  test  of  a  solution  is 
substitution  in  the  given  equations. 


'™^""'''''-"«"'-'"'''i^c^v;m:fr^"^;M;rrri"Nii'7;'^"*'Yiv 


Art.  174]  EXERCISES  20 1 

Example  1.     Solve  the  system 

l2  +  X)/  =  12,  (1) 

y^  +xy  =  4.  (2) 
Solution: 

Adding  (1)  and  (2),  x^  +  2xy  +  if  ^  16,  (3) 

whence  x  +  i/  =  +  4,  or  -  4.  (4) 

Subtracting  (2)  from  (1),  x^  -  y^  =  8.  (5) 

Dividing  (5)  by  (4),  x  -  y  =  +  2,  or  -  2.  (6) 

From  (4)  and  (6),  x  =  3,  or  -3  ;   (/  =  1,  or  -1. 

By  substitution  in  (1)  and  (2)  we  find  that  the  two  pairs  <      ~ 

and  f  ^  "  "^  satisfy  (1)  and  (2). 
ly  =  -1 

Example  2.     Solve  the  system 

x^^f=\%  0) 

t  =  6x.  (2) 

Solution:  Substituting  6x  for  xf  in  (1),  we  obtain 
x2  +  6x  =  16, 
or  x2  +  6x  -  16  =  0.  (3) 

Solving  (3)  by  formula  (Art.  164), 


-6±V3^T64^       ^^_g_ 


2 
If  X  =2,  we  have  y=  ±  \/T2  =  ±  2\/3. 
^      If  X  =  —8,  we  find  y  =±  V—  48,  =  ±  4i-s/3,  which  are  imaginary  numbers. 

8 


The  fKJSsible  solutions  are  then 


ij  =  2\/3,  I  y  =  -2\/3,  1  y  =  4iV3, 


|x=-. 
I  y  =  4i- 


r  X  =  —8 

<  ,  ,  ea(^h  of  which  should  be  tested  by  actual  substitution. 

\t/  =  -4iV3 

EXERCISES 

Solve  and  elieek: 

1.  y-  =  Gx,  3.    X-  +  1/  =  25, 

a;2  -?/-'  +  8  =  0.  xij  =  12. 

2.  a;2  +  ?/2  =  85,  4.   F)X-  -  9//-  +  121  =0, 
a;2  -  2/2  =  77.  7^2  _  3j-2  _  io5  =  0. 


262    EQUATIONS  INVOLVING  QUADRATICS     [Chap.  XXIV. 


yy  -  50  =  0, 
7  =  0. 


5.  x'  +  ?/  =  50, 

9. 

a;2  +  1/2  =  5, 

xy  =  -7. 

x^  -  xy  +  2/2  =  3. 

6.  x^  +  xy  =  36, 

10. 

{x  +  yY  +  ix-y 

xy  +  y^  =  45. 

(x  -  y){x  + 

7.   x'  +  xy  =  10, 

11. 

.T  +  2/  =  5, 

xy  +  ?/  =  6.' 

x?/  =  6. 

8.   p^  -q^  =  9, 

12. 

x'-  -y'  =  9, 

4p^  =  2bq. 

x  +  y  =  9. 

Hint:  Divide  the  members  of  one 
equation  by  the  corresponding  mem- 
bers of  the  other. 


13. 

x'  -  7/  =  1, 
X  -  y  =  S. 

16. 

x'-  +  3xy  +  2y2  =  15 
x  +  y  -3. 

14. 

x'-y'  =  25, 
X  +  1/  =  10. 

17. 

2m2  +  mn  -  rf  =  2, 
2m  -  n  =  1. 

15. 

x^  -xy  -  6y'  = 

16, 

18. 

a2  +  7  =  4a6  +  562, 

X  +  2y  = 

16. 

a2  =  1  -  b\ 

MISCELLANEOUS   PROBLEMS 

1.  The  sum  of  two  numbers  is  5  and  the  sum  of  their  squares 
is  14|.     Find  the  numbers. 

2.  A  rectangular  field  68,200  square  feet  in  area,  is  sur- 
rounded by  a  4-wire  fence  ;  5840  feet  of  wire  were  required 
for  the  fence.     Find  the  dimensions  of  the  field. 

3.  The  sum  of  the  squares  of  two  numbers  is  |^,  and  their 
product  minus  |  is  equal  to  their  difference.  What  are  the 
numbers? 

4.  The  hypotenuse  of  a  right  triangle  is  25  feet.  The  sum 
of  the  other  two  sides  is  35  feet.  What  are  the  lengths  of  the 
sides? 

5.  The  area  of  a  rectangle  is  50  and  the  perimeter  is  54. 
What  are  the  dimensions  of  the  rectangle? 


Art.  174]  PROBLEMS  263 

6.  A  piece  of  wire  48  inclies  long  is  bent  into  the  form  of 
a  right  triangle  in  which  the  hypotenuse  is  20  inches  long.  Find 
the  other  sides  of  the  triangle. 

7.  A  group  of  students  club  together  to  rent  a  suite  of 
rooms  for  $400  per  year.  By  adding  5  new  members  to  the 
group  the  assessment  was  S4  less  per  member.  How  many 
students  were  there  in  the  club  at  first? 

8.  A  certain  number  of  two  figures,  when  multiplied  by 
the  left  digit,  becomes  54;  but  when  multiplied  by  the  right 
digit,  it  becomes  189.     What  is  the  number? 

9.  The  annual  income  from  an  investment  is  S72.  If  the 
principal  were  $240  more  and  the  rate  of  interest  1%  less  the 
income  would  remain  unchanged.  What  are  the  principal  and 
the  rate  of  interest? 

10.  A  sum  of  money  placed  at  simple  interest  for  6  years 
amounts  to  $5580.  Had  the  interest  been  increased  1%  it 
would  have  amounted  to  $45  more  than  this  in  5  years.  What 
are  the  principal  and  the  rate  of  interest? 

11.  A  rectangular  field  is  145  yards  long  and  50  yards 
wide.  How  much  must  the  width  be  increased  and  the  length 
decreased  in  order  that  the  area  remain  unchanged  while  the 
perimeter  is  decreased  30  yards? 

12.  A  certain  kind  of  cloth  loses  5%  in  width  and  4%  in 
length  by  shrinking.  Find  the  length  and  width  of  a  rec- 
tangular piece  of  the  cloth  whose  shrinkage  in  area  is  2.2  square 
yards  and  in  perimeter  2.1  yards? 

13.  An  8  by  10  photograph  is  enlarged  until  it  covers  twice 
the  original  area,  keeping  the  ratio  of  the  length  to  the  width 
unchanged.     Find  the  sides  of  the  enlarged  photograph. 

14.  A  man  loaned  S9000  in  two  unequal  sums  at  such  rates 
that  both  sums  yielded  the  same  annual  interest.  The  larger 
sum  at  the  higher  rate  of  interest  would  yield  S250  per  year, 
and  the  smaller  sum  at  the  lower  rate,  SKiO  per  year.  How 
was  the  money  divided  and  what  were  the  rates  of  interest? 


264      EQUATIONS  INVOLVING  QUADRATICS    [Chap.  XXIV. 

15.  A  dealer  sells  a  number  of  books  for  $1125,  receiving 
the  same  price  for  each  book.  If  he  had  sold  150  books  less 
but  charged  25  cents  more,  he  would  have  received  the  same 
sum.     Find  the  price  and  the  number  of  books. 

16.  A  dealer  purchased  a  certain  number  of  sheep  for 
$175  ;  after  losing  two  of  them  he  sold  the  rest  at  $2.50  a  head 
more  than  he  gave  for  them,  and  by  so  doing  gained  $5  by  the 
deal.     Find  the  number  of  sheep  purchased. 

17.  Both  the  numerator  and  the  denominator  of  a  frac- 
tion are  increased  by  their  squares.  The  new  fraction  reduces 
to  |.  If  both  the  numerator  and  the  denominator  are  divided 
by  their  squares  the  result  is  |.     What  is  the  fraction? 

18.  The  differences  between  the  hypotenuse  of  a  right  tri- 
angle and  the  other  two  sides  are  |  and  1  respectively.  Find 
the  sides  of  the  triangle. 

19.  The  area  and  the  perimeter  of  a  rectangle  are  both 
25.     What  are  the  dimensions  of  the  rectangle? 

20.  A  farmer  is  cutting  wheat  in  a  field  which  is  twice  as 
long  as  it. is  wide.  After  cutting  a  strip  10  rods  wide  around 
the  outside  of  the  field  he  estimates  that  ^  of  the  work  has  been 
done.     What  are  the  dimensions  of  the  field? 


i;i:aHSa:iK!i!!lik'??'?!!:(SSiK"35Si.^StH!:^ISriSKr^^ 


Akt.  174] 


EXERCISES  AND  PROBLEMS 


265 


REVIEW   EXERCISES   AND    PROBLEMS 
Extract  the  following  square  roots  : 


1.    Vc^*,     V25a«6'V,     v/one,     V9  +  16,     v'"*  +  2a^b  +  6^. 


V(j2+y)2-4(x*+l/)  +  4. 


2        /4        Mn"  -  20»'  +25        /16  •  100  • 

■    yO     Y         (n  +3)2       '  Y         iihti' 

3.  Simplify:   V^,     Vs^',     \/|'     l/|.     i/^.     \/45^. 

4.  If  s  is  the  side  of  a  triangle  having  equal  sides,  what  is  the  altitude? 
What  is  the  area? 

Solution: 
The  altitude,  h,  bisects  the  base  AC. 


4  "    4 


Then      h^  =  s^  -   (|j   =  s^  - 

and  ^    =  ^/t  =  1^- 

We  have  then  for  the  area. 

Area  =  |  ■  ^V3  =  |\/3. 


6.  The  side  of  a  triangle  having  equal  sides,  is  10  inches.  Find  the 
altitude  and  the  area,  correct  to  two  decimal  places. 

6.  Find  to  two  decimal    places    the   side   of   an   equilateral    triangle 
^     whose  area  is  100  square  inches. 

7.  Solve  the  equation  A  =  irr^  for  r.     Find  r  if  A  =  50  square  inches. 

4  9 

8.  Add T^  and -->  and  express  the  result    as   a   fraction 

2  +  3\/5  4  -  V5 

with  a  rational  denominator. 

9.  Make  up  quadratic  equations  with  the  imknown  r  so  that  the 
values  of  x  shall  be  (a)  positive  integers;  (h)  positive  fractions;  (r)  n(>gative 
integers  ;  (d)  negative  fractions  ;  (c)  both  zero  ;  (/)  one  zero,  and  one  a 
positive  integer. 

10.  Solve  for  x  the  equation  2j-'  +  5x  -  4  =  0.     Calculate  the  roots  to 
two  decimal  places. 

11.  If  the  diameter  of  a  circle  is  increa.sed  by  3  feet  the  area  is  doubled. 
Find  the  diameter,  correct  to  two  decimal  places. 


266     EQUATIONS  INVOLVING   QUADRATICS    [Chap.  XXIV. 

12.  The  perimeter  of  a  rectangle  is  82  inches.  The  diagonal  is  29  inches. 
How  long  are  the  sides? 

13.  The  hypotenuse  of  a  right  triangle  is  20  inches.  If  the  altitude  is 
multiplied  by  \/2  and  the  base  by  "s/S,  the  hypotenuse  is  multiplied  by  |. 
Find  the  base  and  the  altitude. 

14.  Simplify 


ab  +by  i"'  ^  a^  +by 


a  b\ 

-b) 


2x+y  =4, 

^'''  \2x-hy  =3. 

2x  +y  =  10; 

a  +b 

16.    What  number  added  to  the  denominators  of  ^  and  ^  respectively 

will  make  the  results  equal?     Under  what  condition  is  a  solution  impossi- 
ble? 

16.  Plot  the  loci  of  the  following  equations  and  determine  the  solution 
of  each  pair  from  the  graphs  : 

17.  Solve  the  equation  s  =  ut  +  \JP  for  t.  Find  t  when  u  =  10,/  =  32.2, 
and  s  =  200. 

18.  Plot  the  locus  of  the  equation  y  =  x^  for  values  of  x  from  1  to  10. 
Then  solve  the  equation  for  x.  This  gives  x  =  s/y.  If  now  we  choose  a 
value  of  y  to  be  4,  and  read  off  from  the  graph  the  corresponding  value 
of  X,  which  is  2,  we  have  the  square  root  of  the  chosen  value  of  y.  Thus 
determine  from  the  graph,  as  accurately  as  possible,  the  square  root  of  each 
of  the  following  :   16,  25,  36,  7,  2,  3,  60,  42. 

19.  Which  is  the  larger  V3  or  ^? 

20.  Find  the  values  of  — ;= —  to  three  significant  figiu-es  (1)  by 

V7  -  V5 
using  the  square  roots  of  5,  7,  and  11  ;   (2)  by  first  rationalizing  the  denom- 
inator. 

7 

21.  Evaluate  to  three  significant  figures  — = 


V7-2 


INDEX 


[The  numbers  refer  to  pages.] 


Abscissa,  180 

Absolute  value,  25 

Addition, 

associative  law  for,  56 
commutative  law  for,  56 
of  fractions,  143 
of  monomials,  54 
of  polj-nomials,  57 
of  radicals,  229 
of  signed  numbers,  28 
on  the  number  scale,  23 

Ahtnes,  80,  137 

Algebraic  expressions,  14 

Antecedent,  169 

Arabic  notation,  13 

Axes  of  coordinates,  180 

Base  of  a  power,  9 
Binomial,  53 

cube  of,  105 

square  of,  100 
Braces,  15 
Brackets,  15 

Cancellation,  141 
Checking, 

an  operation,  57 

a  solution,  45 
Circle, 

area,  7 

circumference,  7 
Coefficient,  9 

Completing  the  square,  239,  240 
Complex  fractions,  151 


Consequent,  169 
Constant,  175 
Coordinates,  181 
Cw6e, 

of  a  binomial,  105 

of  a  number,  10 

Denominator,  39,  136 

rationalization  of,  233 
Descartes,  27,  80,  181 
Difference, 

of  two  cubes,  119 

of  two  squares,  113 
Distributive  law  for,  73 
Dividend,  38 
Division,  38,  86 

by  zero,  138 

law  of  exponents  for,  86 

of  fractions,  149 

of  monomials,  86 

of  polynomials,  87,  89 

of  quadratic  surds,  232 

of  signed  numbers,  38 

rule  of  exponents  for,  86 

rule  of  signs  for,  38 
Divisor,  38 

greatest  common,  132 

Elimination,  194 

by    addition    and   subtraction, 
195 

by  substitution,  197 
Eqiuditij,  42 

members  of  an,  42 


268 


INDEX 


Equation,  43 

clearing  of  fractions,  159 

graph  of  an,  186 

historical  note  on,  80 

involving  fractions,  81,  159 

involving  parentheses,  78 

involving  radicals,  235 

linear  in   two  unknowns,  187 

linear  or  simple,  96 

literal,  164,  205 

locus  of,  186,  187 

principles  used  in  solving,  44 

quadratic,  126 

solution  or  root  of,  44 

solved  by   factoring,  126,  127, 
238 
Equations, 

dependent  or  equivalent,  193 

graphical  solution  of,  187 

inconsistent,  193 

independent,  193 

simultaneous,  193 

solution  of  simultaneous  linear, 
188,  193 

system  of  linear,  194 

system  of  quadratic,  256 
Evaluation 

of  expressions,  16 
Exponent,  9 
Exponents, 

law  for  division,  86 

law  for  multiplication,  69 
Expression,  14 

terms  of  an,  53 
Extremes, 

of  a  proportion,  170 

Factor,  9,  109,  110 
common,  132 
found  by  grouping,  112 
highest  common,  132 
integral,  110 


Factor,  monomial,  110,  111 

of  trinomials,  116,  118 

prime,  109,  110 

rational  and  integral,  110 

rationalizing,  234 
Factoring,  109 

equations  solved  by,  126,  127, 
238 

summary  of,  121 
Fractions,  39,  136 

addition    and    subtraction    of, 
143 

clearing  equations  of,  159 

complex,  151 

division  of,  149 

equations  involving,  81 

historical  note  on,  137 

lowest  terms  of,  140 

multipHcation  of,  145 

reduction  to  common  denom- 
inator, 142 

reduction  to  lowest  terms,  140 

signs  in,  138 

square  root  of,  227 

terms  of,  39,  136 
Function,  183 

graph  of,  184,  247 


G7-aph, 

of  a  function,  184,  247 

of  an  equation,  186 
Graphical  meaning, 

of  imaginary  roots,  250  m 

Graphical  representation,  19,  180  ^ 

historical  note  on,  181 

of  positive  and  negative  num- 
bers, 23,  24 

of  scientific  data,  189 
Graphical  solution, 

of  equations,  187 
Greater  than,  25 


INDEX 


269 


Highest  common  factor,  132 
Historical  note, 

on  fractions,  137 

on  graphical  representation,  181 

on  negative  numbers,  27 

on  symbols,  12 

on  the  equation,  SO 

Idvntily,  42 

Imaginary  numbers,  249 
Index  of  a  root,  220 
Integral  expression,  110 
Irrational  7iumbers,  226 

Less  than,  25 
Linear  equation.^, 

in  one  unknown,  96 

in  three  or  more  unknowns,  20(5 

in  two  unknowns,  187 

locus  of,  187 

standard  form  in  two  unknowns, 
199 

systems  of,  194,  206 
Locus, 

of  a  linear  equation,  187 

of  an  equation,  186 
Lowest  common  multiple,  134 

Mean  proportional,  171 
Means, 

of  a  proportion,  170 
Members  of  an  eqiudiiy,  42 
Monomials,  53 

addition  of,  54 

division  of,  86 

factors  of,  110 

product  of,  70 

square  roots  of,  214 

subtraction  of,  60 
Multiple, 

common,  134 

lowest  common,  134 


commutative 


Multiplication,  69 
associative    and 

laws  of,  70 
by  a  monomial,  72 
distributive  law  of,  73 
in  arithmetic,  35 
law  of  exponents  for,  69 
of  a  product,  72 
of  fractions,  145 
of  monomials,  70 
of  polynomials,  72,  75 
of  powers,  69 
of  quadratic  surds,  231 
of  signed  numbers,  35 
rule  of  signs  for,  36 
signs  of,  2 

Negative  numbers,  24 

historical  note  on,  27 
Numbers, 

graphical    representations 
23,  24 

imaginary,  249 

irrational,  226 

positive  and  negative,  24 

rational,  226 

signed,  25 
Numerals, 

Arabic,  13 

Roman,  13 
Numerator,  39,  136 
Numerical  value,  25 


of, 


Order  of  operations 
Ordinate,  180 
Origin  of  coordinates, 


14 


180 


Parentheses, 

equations  involving,  78 
forms  of,  15 
insertion  of,  66 
removal  of,  64 
uses  of,  15 


270 


INDEX 


Polynomials,  53 

addition  of,  57 

arrangement  according  to  as- 
cending and  descending 
powers,  57 

division  of,  87,  89 

factors  of,  111 

multiplication  of,  72,  75 

simplifying,  56 

subtraction  of,  61 
Positive  numbers,  24 
Power,  9 
Powers, 

ascending  and  descending,  57 

product  of,  69 
Prime, 

expressions  prime  to  each  other, 
132 

factor,  109J  110 
Product,  35 

cross,  118 

of  a  polynomial  by  a  mono- 
mial, 72 

of  monomials,  70 

of  polynomials,  75 

of  powers,  69 

of  the  sum  and  difference,  102 

of  two  binomials  with  a  com- 
mon term,  104 
Produx:ts, 

important  type,  100 
Proportion,  170 

by  addition,  173 

by  addition  and  subtraction, 
174 

by  alternation,  173 

by  composition,  173 

by  composition  and  division 
174 

by  division,   174 

by  inversion,  173 

by  subtraction,  174 


Proportional, 
fourth,  172 
mean,  171 
third,  171 


Quadratic  equations,  126,  127,  238 

in  two  unknowns,  256 

solved     by      completing     the 
square,  239 

solved  by  factoring,  127,  238 

solved  by  formula,  242 

special  forms  of,  245 

type  form  of,  242 
Quadratic  surd,  227 

division  of,  232 

multipUcation  of,  231 
Quadratic  trinomial,  118 
Quotient,  38 

of  monomials,  86 

of  polynomials,  87,  89 

Radicals,  226 

addition    and    subtraction    of, 
229 

division  of,  232 

equations  involving,  235 

multiplication  of,  231 

similar,  229 

simplification  of,  227 
Radical  sign,  214 
Radicand,  228 
Ratio,  169 
Rational, 

expression,  109,  110 

number,  226 
Rationalization, 

of  the  denominator,  233 
Rationalizing  factor,  234 
Reciprocal,  153 
Reduction, 

of  fractions  to  lowest  terms,  140 

to  common  denominator,  142 


INDEX 


271 


Right  triangle,  80 
Roman  numerals,  13 

Scale, 

for  representing   numbers,   23 
Signed  numbers,  25 

addition  of,  28 

division  of,  38 

multiplication  of,  35 

subtraction  of,  29 
Signs, 

in  fractions,  138 

rule  of  for  division,  38 

rule  of  for  multiplication,  36 
Similar  triangles,  178 
Simultaneous  equations,  193 

of  second  degree,  256 
Solution, 

of  an  equation,  44 

of  a  pair  of  linear  equations, 
188,  192 

of  simultaneous  quadratics,  257 
Squ/ire, 

completing  the,  239 

of  a  binomial,  100 

of  a  number,  10 

of  a  trinomial,  106 

trinomial,  114 
Square  root,  214 

of  decimals,  221 

of  fractions,  227 

of  monomials,  214 

of  numbers  expressed  in  Arabic 
figures,  219 

of  polynomials,  217 

of  trinomials,  216 

process  of  finding  a,  216 
Subscripts,  165 
Subtraction,  29 

of  fractions,  143 


of  monomials,  60 

of  polynomials,  61 

of  radicals,  229 

of  signed  numbers,  29 

on  the  number  scale,  23,  30 

rules  for,  31 
Sum, 

of  signed  numbers,  28 

of  two  cubes,  119 
Surd,  226 

division  of,  232 

multipUcation  of,  231 

quadratic,  227 
Sy?nbols  of  operation,  1 

historical  note  on,  12 
System, 

of   equations   invoh-ing   quad- 
ratics, 256 

of  linear  equations,  194 

Term,  53 

similar  or  like,  53 
Transposition,  46 
Trinomial,  53 

general  quadratic,  118 

square,  114 

square  of  a,  106 

square  root  of  a,  216 

Unknown,  44 

Variable,  175,  183 
Variation,  175 
Vinculum,  15 


Zero, 


division  by,  138 
origin  of,  13 


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